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All Textbook Solutions for University Physics Volume 1
An elevator filled with passengers has a mass of 1.70103kg . (a) The elevator accelerates upward from rest at a rate of 1.20m/s2for 1.50 s. Calculate the tension in the cable supporting the elevator. (b) The elevator continues upward at constant velocity for 8.50 s. What is the tension in the cable during this time? (c) The elevator decelerates at a rate of 0.600 in/s2 for 0.600m/s2 3.00 s. What is the tension in the cable during deceleration? (d) How high has the elevator moved above its original starting point, and what is its final velocity?A 20.O-g ball hangs from the roof of a freight car by a string. When the freight car begins to move, the string makes an angle of 35.0with the vertical. (a) What is the acceleration of the freight car? (b) What is the tension in the string?A student’s backpack, full of textbooks, is hung from a spring scale attached to the ceiling of an elevator. When the elevator is accelerating downward at 3.8m/s2 , the scale reads 60 N. (a) What Is the mass of the backpack? (b) What does the scale read if the elevator moves upward while speeding up at a rate 3.8m/s2 ? (c) What does the scale read if the elevator moves upward at constant velocity? (d) If the elevator had no brakes and the cable supporting it were to break loose so that the elevator could fall freely, what would the spring scale read?A service elevator takes a load of garbage, mass 10.0 kg, from a floor of a skyscraper under construction, down to ground level, accelerating downward at a rate of 1.2m/s2 . Find the magnitude of the force the garbage exerts on the floor of the service elevator?A roller coaster car starts from rest at the top of a track 30.0 m long and inclined at 20.0to the horizontal. Assume that friction can be ignored. (a) What is the acceleration of the car? (b) How much time elapses before it reaches the bottom of the track?The device shown below is the Atwood’s machine considered in Example 6.5. Assuming that the masses of the string and the frictionless pulley are negligible, (a) find an equation for the acceleration of the two blocks; (b) find an equation for the tension in the string; and (c) find both the acceleration and tension when block 1 has mass 2.00 kg and block 2 has mass 4.00 kg.Two blocks are connected by a massless rope as shown below. The mass of the block on the table is 4.0 kg and the hanging mass is 1.0 kg. The table and tI pulley are frictionless. (a) Find the acceleration of the system. (b) Find the tension in the rope. (c) Find the speed with which the hanging mass hits the floor if it starts from rest and is initially located 1.0 m from the floor.Shown below are two carts connected by a cord that passes over a small frictionless pulley. Each cart rolls freely with negligible friction. Calculate the acceleration of the carts and the tension in the cord.A 2.00 kg block (mass 1) and a 4.00 kg block (mass 2) are connected by a light sting as shown; the inclination of the ramp is 40.0 . Friction is negligible. What is (a) the acceleration of each block and (b) the tension in the sting?Friction (a) When rebuilding his car’s engine, a physics major must exert 3.00102Nof force to insert a dry steel piston into a steel cylinder What is the normal force between the piston and cylinder? (b) What force would he have so exert if the steel pain were oiled?(a) What is the maximum frictional force in the knee joint of a person who supports 66.0 kg of her mass on that knee? (b) During strenuous exercise, it is possible to exert forces to the joints that are easily 10 times greater than the weight being supported. What is the maximum force of friction under such conditions? The frictional forces in joints are relatively small in all circumstances except when the joints deteriorate, such as from injury or arthritis. Increased frictional forces can cause further damage and pam.Suppose you have a 120-kg wooden crate resting on a wood floor, with coefficient of static friction 0.500 between these wood surfaces. (a) What maximum force can you exert horizontally on the crate without moving it? (b) If you continue to exert this force once the crate starts to slip, what will its acceleration then be? The coefficient of sliding friction is known to be 0.300 for this situation.(a) If half of the weight of a small 1.00103-kgutility truck is supported by its two drive wheels, s hat is the maximum acceleration it can achieve on dry concrete? (b) Will a metal cabinet lying on the wooden bed of the truck slip if it accelerates at this rate? (c) Solve both problems assuming the truck has four-steel drive.A team of eight dogs pulls a sled with waxed wood runners on wet snow (mush!). The dogs have average masses of 19.0 kg, and the loaded sled with its rider has a mass of 210 kg. (a) Calculate the acceleration of the dogs starting from rest If each dog exerts an average force of 185 N backward on the snow. (b) Calculate the force in the coupling between the dogs and the sled.Consider the 65.0-kg ice skater being pushed by two others shown below. (a) Find the direction and magnitude of Ftot . the total force exerted on her by the others, given that the magnitudes F1and F2are 26.4 N and 18.6 N, respectively. (b) What is her initial acceleration if she is initially stationary and wearing steel-bladed skates that point in the direction of Ftot ? (c) What is her acceleration assuming she is already moving in the direction of Ftot ? (Remember that friction always acts in the direction opposite that of motion or attempted motion between surfaces in contact)Show that the acceleration of any object down a frictionless incline that makes an angle with the horizontal is a=gsin . (Note that this acceleration is independent of mass.)Show that the acceleration of any object down an incline where friction behaves simply (that Is, where fk=kN ) is a=(sinkcos) . Note that the acceleration Is independent of mass and reduces to the expression found in the previous problem when friction becomes negligibly small (k=0) .Calculate the deceleration of a snow boarder going up a 5.00slope, assuming the coefficient of friction for waxed wood on wet snow The result of the preceding problem may be useful, but be careful to consider the fact that the snow boarder is going uphill.A machine at a post office sends packages out a chute and down a ramp to be loaded into delivery vehicles. (a) Calculate the acceleration of a box heading down a 10.0slope, assuming the coefficient of friction for a parcel on waxed wood is 0.100. (b) Find the angle of the slope down which this box could move at a constant velocity. You can neglect air resistance in both parts.If an object is to rest o an incline without slipping, then friction must equal the component of the weight of the object parallel to the incline. This requires greater and greater friction for steeper slopes. Show that the maximum angle of an incline above the horizontal for which an object will not slide down is =tan1s . You may use the result of the previous problem. Assume that a=0and that static friction has reached its maximum value.Calculate the maximum acceleration of a car that is heading down a 6.00slope (one that makes an angle of 6.00with the horizontal) under the following road conditions. You may assume that the weight of the car is evenly distributed on all four tires and that the coefficient of static friction is involved—hat Is, the tires are not allowed to slip during the deceleration. (Ignore rolling.) Calculate for a car (a) On dry concrete. (b) On wet concrete. (c) On ice, assuming that s=0.100 , the same as for shoes on ice.Calculate the maximum acceleration of a car that is heading up a 4.00slope (one that makes an angle of 4.00with the horizontal) under the following road conditions. Assume that only half the weight of the car is supported by the two drive wheels and that the coefficient of static friction is involved—that is, the tires are not allowed to slip during the acceleration. (Ignore rolling.) (a) On wet concrete. (b) On wet concrete. (C) On ice, assuming that s=0.100 , the same as for shoes on ice.Repeat the preceding problem for a car with four- wheel drive.A freight train consists of two 8.00105kgengines and 45 cars with average masses of 5.50105kg . (a) What force must each engine exert backward on the track to accelerate the train at a rate of 5.0010-2m/s2if the force of friction is 7.50105Nassuming the engines exert identical forces? This is not a large frictional force for such a massive system. Rolling friction for trains is small, and consequently, trains are very energy-efficient transportation systems. (b) What is the force in the coupling between the 37th and 38th cars (this is the force each exerts on the other), assuming all cars have the same mass and that friction is evenly distributed among all of the cars and engines?Consider the 52.0-kg mountain climber shown below. (a) Find the tension in the rope and the force that the mountain climber must exert with her feet on the vertical rock face to remain stationary. Assume that the force is exerted parallel to her legs. Also, assume negligible force exerted by her arms. (b) What is the minimum coefficient of friction between her shoes and the cliff?A contestant in a winter sporting event pushes a 45.0-kg block of ice across a frozen lake as shown below. (a) Calculate the minimum force Fhe must exert to get the block moving. (b) What is its acceleration once it starts to move, if that force is maintained?The contestant now pulls the block of ice with a rope over his shoulder at the same angle above the horizontal as shown below. Calculate the minimum force Fhe must exert to get the block moving. (b) What is its acceleration once it starts to move, if that force is maintained?At a post office, a parcel that is a 20.0-kg box slides do a ramp inclined at 30.0with the horizontal. The coefficient of kinetic friction between the box and plane is 0.0300. (a) Find the acceleration of the box. (b) Find the velocity of the box as it reaches the end of the plane, if the length of the plane is 2 m and the box starts at rest.(a) A 22.0-kg child is riding a playground merry-go-round that is rotating at 40.0 rev/mm. What centripetal force is exerted if he is 1.25 m from its center? (b) What centripetal force is exerted if the merry-go-round rotates at 3.00 rev/min and he Is 8.00 m from its center? (c) Compare each force with his weight.Calculate the centripetal force on the end of a 100-m(radius) wind turbine blade that Is rotating at 0.5 rev/s. Assume the mass is 4 kg.What Is the ideal banking angle for a gentle turn of 1.20-km radius on a highway with a 105 km/h speed limit (about 65 mi/h), assuming everyone travels at the limit?What is the ideal speed to take a 100.0-m-radius curve banked at a 20.0angle?(a) What is the radius of a bobsled turn banked at 75.0and taken at 30.0 m/s, assuming it is ideally banked? (b) Calculate the centripetal acceleration. (c) Does this acceleration seem large to you?Part of riding a bicycle involves leaning at the correct angle when making a turn, as seen below. To be stable, the force exerted by the ground must be on a line going through the center of gravity. The force on the bicycle wheel can be resolved into two perpendicular components—friction parallel to the road (this must supply the centripetal force) and the vertical normal force (which must equal the system’s weight). (a) Show that (as defined as shown) is related to the speed vand radius of curvature rof the turn in the same way as for an ideally banked roadway—that is, =tan1(v2/rg) . (b) Calculate for a 12.0-m/s turn of radius 30.0 m (as in a race).If a car takes a banked curve at less than the ideal speed, friction is needed to keep It from sliding toward the inside of the curve (a problem on icy mountain roads). (a) Calculate the ideal speed to take a 100.0 m radius curve banked at 15.0. (b) What is the minimum coefficient of friction needed for a frightened driver to take the same curve at 20.0 km/h?Modem roller coasters have vertical loops like the one shown here. The radius of curvature is smaller at the top than on the sides so that the downward centripetal acceleration at the top will be greater than the acceleration due to gravity, keeping the passengers pressed firmly into their seats. (a) What is the speed of the roller coaster at the top of the loop if the radius of curvature there is 15.0 m and the downward acceleration of the car is 1.50 g ? (b) How high above the top of the loop must the roller coaster start from rest, assuming negligible friction? (c) If it actually starts 5.00 m higher than your answer to (b), how much energy did it lose to friction? Its mass is 1.50103kg .A child of mass 40.0 kg is in a roller coaster car that travels in a loop of radius 7.00 m. At point A the speed of the car is 10.0 m/s, and at point B, the speed is 10.5 m/s. Assume the child is not holding on and does not wear a seat belt. (a) What is the force of the car seat on the child at point A? (b) What is the force of the car seat on the child at point B? (c) What minimum speed is required to keep the child in his seat at point A?In the simple Bohr model of the ground state of the hydrogen atom, the electron travels in a circular orbit around a fixed proton. The radius of the orbit is 5.281011m , and the speed of the electron is 2.18106m/s . The mass of an electron is 9.111031kg . What is the force on the electron?Railroad tracks follow a circular curve of radius 500.0 m and are banked at an angle of 5.0 . For trains of what speed are these tracks designed?The CERN particle accelerator is circular with a circumference of 7.0 km. (a) What is the acceleration of the protons (m=1.671027kg)that move around the accelerator at 5of the speed of light? (The speed of light is v=3.00108m/s .) (b) What is the force on the protons?A car rounds an unbanked curve of radius 65 m. If the coefficient of static friction between the road and car is 0.70, what is the maximum speed at which the car can traverse the curve without slipping?A banked highway is designed for traffic moving at 90.0 km/h. The radius of the curve is 310 m. What is the angle of banking of the highway?Drag Force and Terminal Speed The terminal velocity of a person falling in air depends upon the weight and the area of the person facing the fluid. Find the terminal velocity (in meters per second and kilometers per hour) of an 80.0-kg skydiver falling in a pike (headfirst) position with a surface area of 0.140m2 .A 60.0-kg and a 90.0-kg skydiver jump from an airplane at an altitude of 6.00103m , both falling in the pike position. Make some assumption on their frontal areas and calculate their terminal velocities. How long will it take for each skydiver to reach the ground (assuming the time to reach terminal velocity is small)? Assume all values are accurate to three significant digits.A 560-g squirrel with a surface area of 930cm2falls from a 5.0-m tree to the ground. Estimate its terminal velocity. (Use a drag coefficient for a horizontal skydiver.) What will be the velocity of a 56-kg person hitting the ground, assuming no drag contribution in such a short distance?To maintain a constant speed, the force provided by a car’s engine must equal the drag force plus the force of friction of the road (the rolling resistance). (a) What are the drag forces at 70 km/h and 100 km/h for a Toyota Camry? (Drag area is 0.70m2 ) (b) What is the drag force at 70 km/h and 100 km/h for a Hummer H2? (Drag area is 2.44m2 ) Assume all values are accurate to three significant digits.By what factor does the drag force on a car increase as it goes from 65 to 110 km/h?Calculate the velocity a spherical rain drop would achieve falling from 5.00 km (a) in the absence of air drag (b) with air drag. Take the size across of the drop to be 4 mm, the density to be 1.00103kg/m3 , and the surface area to be r2 .Using Stokes’ law, verify that the units for viscosity are kilograms per meter per second.Find the terminal velocity of a spherical bacterium (diameter 2.00m ) falling in water. You will first need to note that the drag force is equal to the weight at terminal velocity. Take the density of the bacterium to be 1.10103kg/m3 .Stokes’ law describes sedimentation of particles in liquids and can be used to measure viscosity. Particles in liquids achieve terminal velocity quickly. One can measure the time it takes for a particle to fall a certain distance and then use Stokes’ law to calculate the viscosity of the liquid. Suppose a steel ball bearing (density 7.8103kg/m3 , diameter 3.0 mm) is dropped in a container of motor oil. It takes 12 s to fall a distance of 0.60 m. Calculate the viscosity of the oil.Suppose that the resistive force of the air on a skydiver can be approximated by f=bv2 . If the terminal velocity of a 50.0-kg skydiver is 60.0 m/s, what is the value of b?A small diamond of mass 10.0 g drops from a swimmers earring and falls through the water, reaching a terminal velocity of 2.0 m/s. (a) Assuming the frictional force on the diamond obeys f=bv2 , what is b ? (b) How far does the diamond fall before it reaches 90 percent of its terminal speed?(a) What is the final velocity of a car originally traveling at 50.0 km/h that decelerates at a rate of 0.400m/s2for 50.0 s? Assume a coefficient of friction of 1.0. (b) What is unreasonable about the result? (c) Which premise is unreasonable, or which premises are inconsistent?A 75.0-kg man stands on a bathroom scale in an elevator that accelerates from rest to 30.0 m/s in 2.00 s. (a) Calculate the scale reading in newtons and compare it with her weight. (The scale exerts an upward force on her equal to its reading.) (b) What Is unreasonable about the result? (c) Which premise is unreasonable, or which premises are inconsistent?(a) Calculate the minimum coefficient of friction needed for a car to negotiate an unbanked 50.0 m radius curve at 30.0 m/s. (b) What is unreasonable about the result? (c) Which premises are unreasonable or inconsistent?As shown below, if M=5.50kg , what is the tension in string 1?As shown below, if F=60.0Nand M=4.00kg, what is the magnitude of the acceleration of the suspended object? All surfaces are frictionless.As shown below, if M=6.0kg, what is the tension in the connecting string? The pulley and all surfaces are frictionless.A small space probe Is released from a spaceship. The space probe has mass 20.0 kg and contains 90.0 kg of fuel. It starts from rest in deep space, from the origin of a coordinate system based on the spaceship, and bums fuel at the rate of 3.00 kg/s. The engine provides a constant thrust of 120.0 N. (a) Write an expression for the mass of the space probe as a function of time, between 0 and 30 seconds, assuming that the engine ignites fuel beginning a t=0 . (b) What is the velocity after 15.0 s? (c) What is the position of the space probe after 15.0 s, with initial position at the origin? (d) Write an expression for the position as a function of time, for t30.0s .A half-full recycling bin has mass 10 kg and is pushed up a 40.0incline with constant speed under the action of a 26-N force acting up and parallel to the incline. The incline has fiction. What magnitude force must act up and parallel to the incline for the bin to move down the incline at constant velocity?A child has mass 6.0 kg and slides down a 35incline with constant speed under the action of a 34-N force acting up and parallel to the incline. What is the coefficient of kinetic friction between the child and the surface of the incline?The two barges shown here are coupled by a cable of negligible mass. The mass of the front barge is 2.00103kgand the mass of the rear barge is 3.00103kg . A tugboat pulls the front barge with a horizontal force of magnitude 20.0103N . and the frictional forces of the water on the front and rear barges are 8.00103Nand 10.0103N . respectively. Find the horizontal acceleration of the barges and the tension in the connecting cable.If the order of the barges of the preceding exercise is reversed so that the tugboat pulls the 3.00103-kgbarge with a force of 20.0103N , what are the acceleration of the barges and the tension in the coupling cable?An object with mass m moves along the x -axis. Its position at any time is given by x(t)=pt3+qt2where pand qare constants. Find the net force on this object for any time t .A helicopter with mass 2.35104kg has a position given by r(t)=(0.020t3)i +(2.2t)j -(0.060t2)k . Find the net force on the helicopter at t=3.0s .Located at the origin, an electric car of mass mis at rest and in equilibrium. A time dependent force of F(t)is applied at time t=0 , and its components are Fx(t)=p+ntand Fy(t)=qtwhere p , q , and nare constants. Find the position r(t)and velocity v(t)as functions of time t .A particle of mass mis located at the origin. It is at rest and in equilibrium. A time-dependent force of F(t)is applied a time t=0 , and its components are Fx(t)=ptand Fy(t)=n+qtwhere p , q , and nare constants. Find the position r(t)and velocity v(t)as functions of time t .A 2.0-kg object has a velocity of at t=0 . A constant resultant force of then acts on the object for 3.0 s. What is the magnitude of the object’s velocity at the end of the 3.0-s interval?A 1.5-kg mass has an acceleration of (4.0 i 3.0 j )m/s2 . Only two forces act on the mass. If one of the forces is . what is the magnitude of the other force?A box is dropped onto a conveyor belt moving at 3.4 m/s. If the coefficient of friction between the box and the belt is 0.27, how long will it take before the box moves without slipping?Shown below is a 10.0-kg block being pushed by a horizontal force of magnitude 200.0 N.The coefficient of kinetic friction between the two surfaces is 0.50. Find the acceleration of the block.As shown below, the mass of block 1 is m1=4.0kg . while the mass of block 2 is m2=8.0kg . The coefficient of friction between m1and the inclined surface is k=0.40 . What is the acceleration of the system?A student is attempting to move a 30-kg mini-fridge into her dorm room. During a moment of Inattention, the mini-fridge slides down a 35 degree incline at constant speed when she applies a force of 25 N acting up and parallel to the Incline. What is the coefficient of kinetic friction between the fridge and the surface of the incline?A crate of mass 100.0 kg rests on a rough surface inclined at an angle of 37.0with the horizontal. A massless rope to which a force can be applied parallel to the surface is attached to the crate and leads to the top of the incline. In its present state, the crate is just ready to slip and start to move down the plane. The coefficient of friction is 80of that for the static case. (a) What is the coefficient of static friction? (b) What is the maximum force that can be applied upward along the plane on the rope and not move the block? (c) With a slightly greater applied force, the block will slide up the plane. Once it begins to move, what is its acceleration and what reduced force is necessary to keep it moving upward at constant speed? (d) If the block is given a slight nudge to get it started down the plane, what will be its acceleration in that direction? (e) Once the block begins to slide downward, what upward force on the rope is required to keep the block from accelerating downward?A car is moving at high speed along a highway when the driver makes an emergency braking. The wheels become locked (stop rolling), and the resulting skid marks are 32.0 meters long. If the coefficient of kinetic friction between tires and road is 0.550, and the acceleration was constant during braking, how fast was the car going when the wheels became locked?A crate having mass 50.0 kg falls horizontally off the back of the flatbed truck, which is traveling at 100 km/h. Find the value of the coefficient of kinetic friction between the road and crate if the crate slides 50 m on the road in coming to rest. The initial speed of the crate is the same as the truck, 100 km/h.A 15-kg sled is pulled across a horizontal, snow-covered surface by a force appiled to a rope at 30 degrees with the horizontal. The coefficient of kinetic friction between the sled and the snow is 0.20. (a) If the force Is 33 N, what is the horizontal acceleration of the sled? (b) What must the force be in order to pull the sled at constant velocity?A 30.O-g ball at the end of a stung is swung in a vertical circle with a radius of 25.0 m. The rotational velocity Is 200.0 cm/s. Find the tension in the string: (a) at the top of the circle, (b) at the bottom of the circle, and (c) at a distance of 12.5 cm from the center of the circle (r=12.5cm).A particle of mass 0.50 kg starts moves through a circular path in the xy -plane with a position given by r(t)=(4.0cos3t) i +(4.0sin3t) j where r is in meters and t is in seconds. (a) Find the velocity and acceleration vectors as functions of time. (b) Show that the acceleration vector always points toward the center of the circle (and thus represents centripetal acceleration). (c) Find the centripetal force vector as a function of time.A stunt cyclist rides on the interior of a cylinder 12 m in radius. The coefficient of static friction between the tires and the wall is 0.68. Find the value of the minimum speed for the cyclist to perform the stunt.When a body of mass 0.25 kg is attached to a vertical massless spring, it is extended 5.0 cm from its unstretched length of 4.0 cm. The body and spring are placed on a horizontal frictionless surface and rotated about the held end of the spring at 2.0 rev/s. How far is the spring stretched?A piece of bacon starts to slide down the pan when one side of a pan is raised up 5.0 cm. If the length of the pan from pivot to the raising point is 23.5 cm, what is the coefficient of static friction between the pan and the bacon?A plumb bob bangs from the roof of a railroad car. The car rounds a circular track of radius 300.0 m at a speed of 90.0 km/b. At what angle relative to the vertical does the plumb bob hang?An airplane flies at 120.0 m/s and banks at a 30angle. If its mass is 2.50103kg . (a) what is the magnitude of the lift force? (b) what is the radius of the turn?The position of a particle is given by r(t)=A(cost i +sint j ) . where is a constant. (a) Show that the particle moves in a circle of radius A. (b) Calculate dr/dt, and then show that the speed of the particle is a constant A . (c) Determine d2r/dt2and show that a is given by ac=r2. (d) Calculate the centripetal force on the particle. [Hint: For (b) and (c), you will need to use (d/dt)(cost)=sint and (d/dt)(sint)=cott .Two blocks connected by a string are pulled across a horizontal surface by a force applied to one of the blocks, as shown below. The coefficient of kinetic friction between the blocks and the surface is 0.25. If each block has an acceleration of 2.0m/s2to the right, what is the magnitude Fof the applied force?As shown below, the coefficient of kinetic friction between the surface and the larger block is 0.20, and the coefficient of kinetic friction between the surface and the smaller block is 0.30. lf F=10Nand M=1.0kg , what is the tension in the connecting string?In the figure, the coefficient of kinetic friction between the surface and the blocks is k. If M=1.0kg , find an expression for the magnitude of the acceleration of either block (in terms of F , k , and g ).Two blocks are stacked as shown below, and rest on a frictionless surface. There is friction between the two blocks (coefficient of friction ). An external force is applied to the top block at an angle with the horizontal. What is the maximum force Fthat can be applied for the two blocks to move together?A box rests on the (horizontal) back of a truck. The coefficient of static friction between the box and the surface on which it rests is 0.24. What maximum distance can the truck travel (starting from rest and moving horizontally with constant acceleration) in 3.0 s without having the box slide?A double-incline plane is shown below. The coefficient of friction on the left surface Is 0.30, and on the right surface 0.16. Calculate the acceleration of the system.In a later chapter, you will find that the weight of a particle varies with altitude such that w=mgr02r2where r0is the radius of Earth and ris the distance from Earth’s center. If the particle is fired vertically with velocity v0from Earth’s surface, determine its velocity as a function of position r. (Hint: use adr=vdv, the rearrangement mentioned in the text.)A large centrifuge, like the one shown below, is used to expose aspiring astronauts to accelerations similar to those experienced in rocket launches and atmospheric reentries. (a) At what angular velocity is the centripetal acceleration 10g if the rider is 15.0 m from the center of rotation? (b) The rider s cage hangs on a pivot at the end of the arm, allowing it to swing outward during rotation as shown in the bottom accompanying figure. At that angle below the horizontal will the cage hang when the centripetal acceleration is 10g? (Hint: The arm supplies centripetal force and supports the weight of the cage. Draw a free-body diagram of the forces to see what the angle should be.)A car of mass 1000.0 kg is traveling along a level road at 100.0 km/h when its brakes are applied. Calculate the stopping distance if the coefficient of kinetic friction of the tires is 0.500. Neglect air resistance. (Hint: since the distance traveled is of interest rather than the time, xis the desired independent variable and not t . Use the Chain Rule to change the variable: dvdt=dvdxdxdt=vdvdx ).An airplane flying at 200.0 m/s makes a turn that takes 4.0 mm. What bank angle is required? What is the percentage increase in the perceived weight of the passengers?A skydiver is at an altitude of 1520 m. After 10.0 seconds of free fall, he opens his parachute and finds that the air resistance, FD , is given by the formula FD=bv , where bis a constant and vis the velocity. If b=0.750 , and the mass of the skydiver is 82.0 kg, first set up differential equations for the velocity and the position, and then find: (a) the speed of the skydiver when the parachute opens, (b) the distance fallen before the parachute opens, (c) the terminal velocity after the parachute opens (find the limiting velocity), and (d) the time the skydiver is in the air after the parachute opens.In a television commercial, a small, spherical bead of mass 4.00 g is released from rest at t=0in a bottle of liquid shampoo. The terminal speed is observed to be 2.00 cm/s. Find (a) the value of the constant bin the equation v=mgb(1ebt/m), and (b) the value of the resistive force when the bead reaches terminal speed.A boater and motor boat ate at rest on a lake. Together, they have mass 200.0 kg. If the thrust of the motor is a constant force of 40.0 N in the direction of motion, and if the resistive force of the water is numerically equivalent to 2 times the speed vof the boat, set up and solve the differential equation to find: (a) the velocity of the boat at time t; (b) the limiting velocity (the velocity after a long time has passed).Check Your Understanding Can kinetic friction ever be a constant force for all paths?Check Your Understanding Can Earth’s gravity ever be a constant force for all paths?Check Your Understanding Find the work done by the same force in Example 7.4 over a cubic path, y=(0.25m-2)x3 , between the same points A=(0,0) and B=(2m,2m) .Check Your Understanding The spring Example 7.5 is compressed 6 cm from its equilibrium length. (a) Does the spring force do positive or negative work and (b) what is the magnitude?Check Your Understanding (a) A car and a truck each moving with the same kinetic energy. Assume that the truck has more mass than the car. Which has the greater speed? (b) A car truck are each moving with same speed. Which has the greater kinetic energy?Check Your Understanding You are rowing a boat parallel to the banks of a river. Your kinetic energy relative to the banks is less than kinetic energy relative to the water. Are you rowing with or against the current?Check Your Understanding suppose the radius of the loop-the-loop in Example 7.9 is 15 cm and the toy car starts from rest at a height of 45 cm above the bottom. What is its speed at the top of the loop?Check Your Understanding Estimate the power expended by a weightlifter raising a 150-kg barbell 2 m in 3 s.Give an example of something we think of as work in everyday circumstances that is not work in the scientific sense. Is energy transferred or changed in form in your example? If so, explain how this is accomplished without doing work.Give an example of a situation in which there is a force and a displacement, but the force does no work. Explain why it does no work.Describe a situation in 4iich a force is exerted for a long time but does no work. Explain.A body moves in a circle at constant speed. Does the centripetal force that accelerates the body do any work? Explain.Suppose you throw a ball upward and catch it when it returns at the same height. How much work does the gravitational force do on the ball over its entire trip?Why is it more difficult to do sit-ups while on a slant board than on a horizontal surface? (See below.)As a young man, Tarzan climbed up a vine to reach his tree house. As he got older, he decided to build and use a staircase instead. Since the work of the gravitational force mg is path Independent, what did the King of the Apes gain in using stairs?A particle of m has a velocity of . Is its kinetic energy given by ? If not, what is the correct expression?One particle has mass mand a second particle has mass 2m . The second particle is moving with speed vand the first with speed 2v . How do their kinetic energies compare?A person drops a pebble of mass m1from a height h, and it hits the floor with kinetic energy K. The person drops another pebble of mass m2, from a height of 2h , and it hits the floor with the same kinetic energy K. How do the masses of the pebbles compare?The person shown below does work on the lawn mower. Under what conditions would the mower gain energy from the person pushing the mower? Under what conditions would it lose energy?Work done on a system puts energy into it. Work done by a system removes energy from it. Give an example for each statement.Two marbles of masses mand 2mare dropped from a height h. Compare their kinetic energies when they reach the ground.Compare the work required to accelerate a car of mass 2000 kg from 30.0 to 400 km/h with that required for an acceleration from 50.0 to 60.0 km/h.Suppose you are jogging at constant velocity. Are you doing any work on the environment and vice versa?Two forces act to double the speed of a particle, initially moving with kinetic energy of 1 J. One of the forces does 4 J of work. How much work does the other force do?Most electrical appliances are rated in watts. Does this rating depend on how long the appliance is on? (When off, It is a zero-wan device.) Explain in terms of the definition of power.Explain, in terms of the definition of power, why energy consumption is sometimes listed in kilowatt-hours rather than joules. What is the relationship between these two energy units?A spark of static electricity, such as that you might receive from a doorknob on a cold dry day, may carry a few hundred watts of power. Explain why you are not injured by such a spark.Does the work done in lifting an object depend on how fast it is lifted? Does the power expended depend on how fast it is lifted?Can the power expended by a force be negative?How can a 50-W light bulb use more energy than a 1000-W oven?Work How much work does a supermarket checkout attendant do on a can of sip he pushes 0.600 m horizontally with a force of 5.00 N?A 75.0-kg person climbs stairs, gaining 2.50 m in height. Find the work done to accomplish this task.(a) Calculate the work done on a 1500-kg elevator car by its cable to lift it 40.0 m at constant speed, assuming friction averages 100 N. (b) What is the work done on the lift by the gravitational force in this process? (c) What is the total work done on the lift?Suppose a car travels 108 km at a speed of 30.0 m/s, and uses 2.0 gal of gasoline. Only 30 of the gasoline goes into useful work by the force that keeps the car moving at constant speed despite friction. (The energy content of gasoline is about 140 Mi/gal.) (a) What is the magnitude of the force exerted to keep the car moving at constant speed? (b) If the required force is directly proportional to speed, how many gallons will be used to drive 108 km at a speed of 28.0 m/s?Calculate the work done by an 85.0-kg man who pushes a crate 4.00 m up along a ramp that makes an angle of 20.00 with the horizontal (see below). He exerts a force of 500 N on the crate parallel to the ramp and moves at a constant speed. Be certain to include the work he does on the crate and on his body to get up the ramp.How much work is done by the boy pulling his sister 30.0 m in a wagon as shown below? Assume no friction acts on the wagon.A shopper pushes a grocery cart 20.0 m at constant speed on level ground, against a 35.0 N frictional force. He pushes in a direction 25.Oc below the horizontal. (a) What is the work done on the cart by friction? (b) What is the work done on the cart by the gravitational force? (c) What is the work done on the cart by the shopper? (d) Find the force the shopper exerts, using energy considerations. (e) What is the total work done on the cart?Suppose the ski patrol lowers a rescue sled and victim, having a total mass of 90.0 kg, down a 60.00 slope at constant speed, as shown below. The coefficient of friction between the sled and the snow is 0.100. (a) How much work is done by fiction as the sled moves 30.0 m along the hill? (b) How much work is done by the rope on the sled in this distance? (c) What is the work done by the gravitational force on the sled? (d) What is the total work done?A constant 20-N force pushes a small ball in the direction of the force over a distance of 5.0 m. What is the work done by the force?A toy cart is pulled a distance of 6.0 m in a straight line across the floor. The force pulling the cart has a magnitude of 20 N and is directed at 37 above the horizontal. What is the work done by this force?A 5.0-kg box rests on a horizontal surface. The coefficient of kinetic friction between the box and surface is K=0.50 . A horizontal force pulls the box at constant velocity for 10 cm. Find the work done by (a) the applied horizontal force, (b) the frictional force, and (c) the net force.A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (K=0.20) at constant velocity by a force directed 25 above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.Suppose that the sled plus passenger of the preceding problem is pushed 20 m across the snow at constant velocity by a force directed 30 below the horizontal. Calculate (a) the work of the applied force, (b) the work of fiction, and (c) the total work.How much work does the force do on a particle as it moves from toHow much work is done against the gravitationaI force on a 5.0-kg briefcase when it is carried from the ground floor to the roof of the Empire State Building, a vertical climb of 380 m?It takes 500 J of work to compress a spring 10 cm. What is the force constant of the spring?A bungee cord is essentially a very long rubber band that can stretch up to four times its unstretched length. However, its spring constant vanes over its stretch [see Menz, P.G. “The Physics of Bungee Jumping.” The Physics Teacher (November 1993) 31: 483-487]. Take the length of the cord to be along the direction and define the stretch as the length of the cord minus its un-stretched length that is, (see below). Suppose a particular bungee cord has a spring constant, for of and for. (Recall that the of (Recall that the spring constant is the slope of the force versus its stretch (a) What is the tension in the cord when the stretch is 16.7 m (the maximum desired for a given jump)? (b) How much work must be done against the elastic force of the bungee cord to stretch It 16.7 m? Figure 7.16 (credit modification of work by Graeme Churchard)A bungee cord exerts a nonlinear elastic force of magnitude where is the distance the cord is stretched, and How much work, must be done on the cord to stretch it 16.7 m?Engineers desire to model the magnitude of the elastic force of a bungee cord using the equation F(x)=a[x+9m9m( 9m x+9m)2] , where x is the stretch of the cord along its length and a is a constant. If it takes 22.0 kJ of work to stretch the cord by 16.7 m, determine the value of the constant a .A particle moving in the xy -plane is subject to a force F(x,y)=(50Nm2)(xi+yj)( x 2 + y 2 )3/2 , where x and y are in meters. Calculate the work done on the particle by this force, as it moves in a straight line from the point (3 in, 4 m) to the point (8 in, 6 in).A particle moves along a curved path y(x)=(10m){1+cos[0.1m1]x} , from x=0 to x=10m , subject to a tangential force of variable magnitude F(x)=(10N)sin[(0.1m-1)x] . How much work does the force do? (Hint: Consult a table of integrals or use a numerical integration program.)Kinetic Energy Compare the kinetic energy of a 20,000-kg truck moving at 110 km/h with that of an 80.0-kg astronaut in orbit moving at 27,500 km/h.(a) How fast must a 3000-kg elephant move to have the same kinetic energy as a 65.0-kg sprinter running at 10.0 m/s? (b) Discuss how the larger energies needed for the movement of larger animals would relate to metabolic rates.Estimate the kinetic energy of a 90,000-ton aircraft carrier moving at a speed of at 30 knots. You will need to look up the definition of a nautical mile to use in converting the unit for speed, where 1 knot equals 1 nautical mile per hour.Calculate the kinetic energies of (a) a 2000.0-kg automobile moving at 100.0 km/h; (b) an 80.-kg runner sprinting at 10. m/s; and (c) a 9.11031 -kg electron moving at 2.0107m/s .A 5.0-kg body has three times the kinetic energy of an 8.0-kg body. Calculate the ratio of the speeds of these bodies.An 8.0-g bullet has a speed of 800 m/s. (a) What is its kinetic energy? (b) What is its kinetic energy if the speed is halved?(a) Calculate the force needed to bring a 950-kg car to rest from a speed of 90.0 km/h in a distance of 120 m (a fairly typical distance for a non-panic stop). (b) Suppose instead the car hits a concrete abutment at full speed and is brought to a stop in 2.00 m. Calculate the force exerted on the car and compare it with the force found in part (a).A car’s bumper is designed to withstand a 4.0-km/ h (1.1-m/s) collision with an immovable object without damage to the body of the car. The bumper cushions the shock by absorbing the force over a distance. Calculate the magnitude of the average force on a bumper that collapses 0.200 m while bringing a 900-kg car to rest from an initial speed of 1.1 m/s.Boxing gloves are padded to lessen the force of a blow. (a) Calculate the force exerted by a boxing glove on an opponent’s face, if the glove and face compress 7.50 cm dui-Ing a blow In which the 7.00-kg arm and glove are brought to rest from an initial speed of 10.0 m/s. (b) Calculate the force exerted by an identical blow in the gory old days when no gloves were used, and the knuckles and face would compress only 2.00 cm. Assume the change in mass by removing the glove is negligible. (c) Discuss the magnitude of the force with glove on. Does it seem high enough to cause damage even though it is lower than the force with no glove?Using energy considerations, calculate the average force a 60.0-kg sprinter exerts backward on the track to accelerate from 2.00 to 8.00 m/s in a distance of 25.0 m, if he encounters a headwind that exerts an average force of 30.0 N against him.A 5.0-kg box has an acceleration of 2.0m/s2 when it is pulled by a horizontal force across a surface with K=0.50 . Find the work done over a distance of 10 cm by (a) the horizontal force, (b) the frictional force, and (c) the net force. (d) What is the change in kinetic energy of the box?A constant 10-N horizontal force is applied to a 20-kg cart at rest on a level floor. If friction is negligible, what Is the speed of the cart when it has been pushed 8.0 m?In the preceding problem, the 10-N force is applied at an angle of 45 below the horizontal. What is the speed of the cart en it has been pushed 8.0 m?Compare the work required to stop a 100-kg crate sliding at 1.0 m/s and an 8.0-g bullet traveling at 500 m/s.A wagon with its passenger sits at the top of a hill. The wagon is given a slight push and rolls 100 m do a 10 incline to the bottom of the hill. What is the wagon’s speed when it reaches the end of the incline. Assume that the retarding force of friction is negligible.An 8.0-g bullet with a speed of 800 m/s is shot in to a wooden block and penetrates 20 cm before stopping. What is the average forte of the wood on the bullet? Assume the block does not move.A 2.0-kg block starts with a speed of 10 m/s at the bottom of a plane inclined at 37 to the horizontal. The coefficient of sliding friction between the block and plane is k=0.30 . (a) Use the work-energy principle to determine how far the block slides along the plane before momentarily coming to rest. (b) After stopping, the block slides back down the plane. What is its speed when It reaches the bottom? (Hint: For the round trip, only the force of friction does work on the block.)When a 3.0-kg block is pushed against a massless spring of force constant constant 4.5103N/m , the spring is compressed 8.0 cm. The block is released, and it slides 2.0 m (from the point at which it is released) across a horizontal surface before friction stops it. What is the coefficient of kinetic friction between the block and the surface?A small block of mass 200 g starts at rest at A, slides to B where its speed is then slides along the horizontal surface a distance 10 m before coming to rest at C. (See below.) (a) What is the work of friction along the curved surface? (b) What is the coefficient of kinetic friction along the horizontal surface?A small object is placed at the top of an incline that is essentially frictionless. The object slides down the incline onto a rough horizontal surface, where it stops in 5.0 S after traveling 60 m. (a) What Is the speed of the object at the bottom of the incline and its acceleration along the horizontal surface? (b) What is the height of the incline?When released, a 100-g block slides down the path shown below, reaching the bottom with a speed of 4.0 m/s. How much work does the force of friction do?A 0.22LR-caliber bullet like that mentioned in Example 7.10 is fired into a door made of a single thickness of 1-inch pine boards. How fast would the bullet be traveling after it penetrated through the door?A sled stalls from rest at the top of a snow-covered incline that makes a 22 angle with the horizontal. After sliding 75 m down the slope, its speed is 14 m/s. Use the work-energy theorem to calculate the coefficient of kinetic friction between the runners of the sled and the snowy surface.A person in good physical condition can put out 100 W of useful power for several hours at a stretch, perhaps by pedaling a mechanism that drives an electric generator. Neglecting any problems of generator efficiency and practical considerations such as resting time: (a) How many people ‘would it take to nm a 4.O0-kW electric clothes dryer? (b) How many people would it take to replace a large electric power plant that generates 800 MW?What is the cost of operating a 3.00-W electric clock for a year if the cost of electricity is 0.0900 per kWh ?A large household air conditioner may consume 15.0 kW of power. What Is the cost of operating this air conditioner 3.00 h per day for 30.0 d if the cost of electricity is $0.110 per kWh ?(a) What is the average power consumption in watts of an appliance that uses 5.00kWh of energy per day? (b) How many joules of energy does this appliance consume in a year?(a) What is the average useful power output of a person who does 6.00106J of useful work in 8.00 h? (b) Working at this rate, how long will it take this person to lift 2000 kg of bricks 1.50 m to a platform? (Work done to lift his body can be omitted because it is not considered useful output here.)A 500-kg dragster accelerates from rest to a final speed of 110 m/s in 400 m (about a quarter of a mile) and encounters an average frictional force of 1200 N. What is its average power output in watts and horsepower if this takes 7.30s?(a) How long will it take an 850-kg car with a useful power output of 40.0 hp (1 hp equals 746 W) to reach a speed of 15.0 m/s, neglecting friction? (b) How long will this acceleration take if the car also climbs a 3.00-m high hill in the process?(a) Fir the useful power output of an elevator motor that Lifts a 2500-kg load a height of 35.0 m in 12.0 s, if it also increases the speed from rest to 4.00 m/s. Note that the total mass of the counterbalanced system is 10,000 kg—so that only 2500 kg is raised in height, but the full 10,000 kg is accelerated. (b) What does it cost, if electricity is $0.0900 per kWh ?(a) How long would it take a 1.50105kg airplane with engines that produce 100 MW of power to reach a speed of 250 m’s and an altitude of 12.0 km if air resistance were negligible? (b) If it actually takes 900 s, what is the power? (c) Given this power, what is the average force of air resistance if the airplane takes 1200 s? (Hint: You must find the distance the plane travels in 1200 s assuming constant acceleration.)Calculate the power output needed for a 950-kg car to climb a 2.00 slope at a constant 30.0 m/s while encountering wind resistance and friction totaling 600 N.A man of mass 80 kg runs up a flight of stairs 20 m high in 10 s. (a) how much power is used to lift the man?(b) If the man’s body is 25 efficient, how much power does he expend?The man of the preceding problem consumes approximately 1.05107J (2500 food calories) of energy per day in maintaining a constant weight. What is the average power he produces over a day? Compare this with his power production when he runs up the stairs.An electron in a television tube is accelerated uniformly from rest to a speed of 8.4107 m/s over a distance of 2.5 cm. What is the power delivered to the electron at the instant that its displacement is 1.0 cm?Coal is lifted out of a mine a vertical distance of 50 m by an engine that supplies 500 W to a conveyer belt. How much coal per minute can be brought to the surface? Ignore the effects of friction.A girl pulls her 15-kg wagon along a flat sidewalk by applying a 10-N force at 37 to the horizontal. Assume that friction is negligible and that the wagon starts from rest. (a) How much work does the girl do on the wagon in the first 2.0 s. (b) How much instantaneous power does she exert at = 2.0 s?A typical automobile engine has an efficiency of 25 . Suppose that the engine of a 1000-kg automobile has a maximum power output of 140 hp. What is the maximum grade that the automobile can climb at 50 km/h if the frictional retarding force on it is 300 N?When jogging at 13 km/h on a level surface, a 70-kg man uses energy at a rate of approximately 850 W. Using the facts that the “human engine” is approximately 25 efficient, determine the rate at which this man uses energy when jogging up a 5.0 slope at this same speed. Assume that the frictional retarding force is the same in both cases.A cart is pulled a distance D on a flat, horizontal surface by a constant farce F that acts at an angle with the horizontal direction. The other forces on the object during this time are gravity (Fw) , normal forces (FN1) and (FN2) , and rolling frictions Fr1 and Fr2 , as shown below What is the work done by each force?Consider a particle on which several forces act, one of which is known to be constant in time: . As a result, the particle moves along the x-axis from x=0 to x=5 m in some time interval. What is the work done by ?Consider a particle on which several forces act, one of which is known to be constant in time: . As a result, the particle moves along the x-axis from x=0 to x=5 m and then parallel to the y-axis from y=0 to y=6 m. What is the work done by ?Consider a particle on which several forces act, one of which is known to be constant in time: . As a result, the particle moves along a straight path from a Cartesian coordinate of (0m,0m) to (5m,6m) . What is the work done by ?Consider a particle on which a force acts that depends on the position of the particle. This force is given by . Find the work done by this force when the particle moves from the origin to a point 5 meters to the right on the x-axis.A boy pulls a 5-kg cart with a 20-N force at an angle of 30 above the horizontal for a length of time. Over this time frame, the cart moves a distance of 12 m on the horizontal floor. (a) Find the work done on the cart by the boy. (b) What will be the work done by the boy if he pulled with the same force horizontally instead of at an angle of 300 above the horizontal over the same distance?A crate of mass 200 kg is to be bright from a site on the ground floor to a third floor apartment. The workers know that they can either use the elevator first, their slide it along the third floor to the apartment, or first slide the crate to another location marked C below and then take the elevator to the third floor and slide it on the third floor a shorter distance. The trouble is that the third floor is very rough compared to the ground floor. Given that the coefficient of kinetic Motion between the crate and the ground floor is 0.100 and between the crate and the third floor surface is 0.300, find the work needed by the workers for each path shown from A to E. Assume that the force the workers need to do is just enough to slide the crate at constant velocity (zero acceleration). Note: The work by the elevator against the force of gravity is not done by the workers.At hokey puck of mass 0.17 kg is shot across a rough floor with the roughness different at different places, which can be described by a position-dependent coefficient of kinetic friction. For a puck moving along the x-axis, the coefficient of kinetic friction is the following function of x, where x is in m: (x)=0.1+0.05x . Find the work done by the kinetic frictional forte on the hockey puck when it h moved (a) from x=0 to x=2 m, and (b) from x=2 m to x=4 m.A horizontal force of 20 N is required to keep a 5.0 kg box traveling at a constant speed up a frictionless incline for a vertical height change of 3.0 m. (a) What Is the work done by gravity dining this change in height? (b) What Is the work done by the normal force? (c) What is the work done by the horizontal farce?A 7.0-kg box slides along a horizontal frictionless floor at 1.7 m/s and collides with a relatively massless spring that compresses 23 cm before the box comes to a stop. (a) How much kinetic energy does the box have before it collides with the spring? (b) Calculate the work done by the spring. (c) Determine the spring constant of the spring.You are driving your car on a straight road with a coefficient of friction between the tires and the road of 0.55. A large piece of debris falls in front of your view and you immediate slam on the brakes, leaving a skid mark of 30.5 m (100-feet) long before coming to a stop. A policeman sees your car stopped on the road, looks at the skid mark, and gives you a ticket for traveling over the 13.4 m/s (30 mph) speed limit. Should you fight the speeding ticket in court?A crate is being pushed across a rough floor surface. If no force is applied on the crate, the crate will slow down and come to a stop. If the crate of mass 50 kg moving at speed 8 m/s comes to rest in 10 seconds, what is the rate at which the frictional force on the crate takes energy away from the crate?Suppose a horizontal force of 20 N is required to maintain a speed of 8 m/s of a 50 kg crate. (a) What is the power of this force? (b) Note that the acceleration of the crate is zero despite the fact that 20 N force acts on the crate horizontally. What happens to the energy given to the crate as a result of the work done by this 20 N force?Grains from a hopper falls at a rate of 10 kg/s vertically onto a conveyor belt that is moving horizontally at a constant speed of 2 m/s. (a) What force Is needed to keep the conveyor belt moving at the constant velocity? (b) What is the minimum power of the motor driving the conveyor belt?A cyclist in a race must climb a 5 hill at a speed of 8 m/s. If the mass of the bike and the biker together is 80 kg, what must be the power output of the biker to achieve the goal?Shown below is a 40-kg crate that is pushed at constant velocity a distance 8.0 m along a 30 Incline by the horizontal force . The coefficient of kinetic friction between the crate and the incline is k=0.40 . Calculate the work done by (a) the applied force, (b) the frictional force, (c) the gravitational force, and (d) the net force.The surface of the preceding problem is modified so that the coefficient of kinetic friction is decreased. The same horizontal force is applied to the crate, and after being pushed 8.0 m, its speed is 5.0 m/s. How much work is now done by the force of friction? Assume that the crate starts at rest.The force F(x) varies with position, as shown beolow Find the work done by this force on a particle as It moves from x=1.0 m to x=5.0 m.Find the work done by the same force in Example 7.4, between the same points, A=(0,0) and B=(2m,2m) , over a circular arc of radius 2 m, centered at (0,2m) . Evaluate the path integral using Cartesian coordinates. (Hint: You will probably need to consult a table of integrals.)Answer the preceding problem using polar coordinates.Find the work done by the same force in Example 7.4, between the same points, A=(0,0) and B=(2m,2m) , over a circular arc of radius 2 m, centered at (2m,0) . Evaluate the path integral using Cartesian coordinates. (Hint: You will probably need to consult a table of integrals.)Answer the preceding problem using polar coordinates.Constant power P is delivered to a car of mass m by its engine. Show that If air resistance can be ignored, the distance covered In a time t by the car, starting from rest, is given by s=(8P/9m)1/2t3/2 .Suppose that the air resistance a car encounters is independent of its speed. When the car travels at 15 m/s, its engine delivers 20 hp to its wheels. (a) What is the power delivered to the wheels when the car travels at 30 m/s? (b) How much energy does the car use in covering 10 km at 15 m/s? At 30 m/s? Assume that the engine is 25 efficient. (c) Answer the same questions if the force of air resistance is proportional to the speed of the automobile. (d) What do these results, plus your experience with gasoline consumption, tell you about air resistance?Consider a linear spring, as in Figure 7.7(a), with mass M uniformly distributed along its length. The left end of the spring is fixed, but the right end, at the equilibrium position x=0 , is moving with speed v in the x-direction. What is the total kinetic energy of the spring? (Hint: First express the kinetic energy of an infinitesimal element of the spring dm in terms of the total mass, equilibrium length, speed of the right-hand end, and position along the spring; then integrate.)Check Your understanding In Example 8.1 what are the potential energies of the particle at x=1 m and x=2 m with respect to zero at x=1.5 ?verify that the difference of potential energy is still 7 J.Check Your Understanding What are the values of the gravitational potential energy of the hiker at the base, and sea level, with respect to a sea-level zero of potential energy?Check Your Understanding When the length of the spring in Example 8.3 changes from an initial value of 22.0 cm to a final value, the elastic potential energy it contributes changes by 0.0800 J. Find the final length.Check Your Understanding Suppose the mass in Equation 8.6 is doubled while keeping the all other conditions the same. Would the maximum expansion of the spring increase, decrease, or remain the same? Would the speed at point B be larger, smaller, or the same compared to the original mass?Check Your Understanding A two-dimensional, conservative force is zero on the x- and y-axes, and satisfies the condition (dFx/dy)=(dFy/dx)=(4N/m3)xy what is the magnitude of the force at the point x=y=1m ?Check Your Understanding Fend the forces on the particle in Example 8.6 when kinetic energy is 1.0 J at x=0.Check Your Understanding How high above the bottom of its arc is the particle in the simple pendulum above, when its speed is 0.81 m/s?Check Your Understanding You probably recall that, neglecting air resistance, if you throw a projectile straight up, the time ii takes to reach its maximum height equals the time it takes to fall from the maximum height back to the starting height. Suppose you cannot neglect air resistance, as in Example 8.8. Is the time the projectile takes to go up (a) greater than, (b) less than, or (c) equal to the time It takes to come back down? Explain.Check Your Understanding What potential energy U(x) can you substitute in Equation 8.13 that will result in motion with constant velocity of 2 m/s for a particle of mass 1 kg and mechanical energy 1 J?Check Your Understanding Repeat Example 8.10 when the particle’s mechanical energy is +0.25 J.Check Your Understanding Find x(t) for the mass-spring system in Example 8.11 ii the particle starts from x0=0 at t=0. what is the particle’s initial velocity?The kinetic energy of a system must always be positive or zero. Explain whether this is true for the potential energy of a system.The force exerted by a diving board is conservative, provided the internal friction is negligible. Assuming friction is negligible, describe changes in the potential energy of a diving board as a swimmer drives from it, starting just before the swimmer steps on the board until just after his feet leave it.Describe the gravitational potential energy transfers and transformations for a javelin, starting from the point at which an athlete picks up the javelin and ending when the javelin is stuck into the ground after being thrown.A couple of soccer balls of equal mass are kiched off the ground at the same speed but at different angles. Soccer ball A is kicked off at an angle slightly above the horizontak, whereas boll B is kicked slightly below the vertical. How do each of the following compare for ball Aand ball B? (a) The initial kinetic energy and (b) the change in gravitational potential energy from the ground to the highest point? If the energy in part (a) differs from part (b), explain why there is a differenne between the two energies.What is the dominant factor that affects the speed of an object that started from rest down a frictionless incline if the only work done on the object is from gravitational forces?Two people observe a leaf falling from a tree. One person is standing on a ladder and the other is on the ground. If each person were to compare the energy of the leaf observed, would each person find the following to be the same or different for the leaf, from the point where it falls off the tree to when it hits the ground: (a) the kinetic energy of the leaf; (b) the change in gravitational potential energy; (c) the final gravitational potential energy?What is the physical meaning of a non-conservative force?A bottle rocket is shot straight up in the air with a speed 30 m/s. if the air resistance is ignores, the bottle would go up to a height of approximately 46 m. however the rocket goes up to only 35 m before returning to the ground. What happened? Explain, giving only a qualitative response.An external force acts on a particle during a trip from one point to another and back to that same point. This particle is only effected by conservative forces. Does this particle’s kinetic energy and potential change as a result of this trip?When a body slides down an inclined plane, does the work of friction depend on the body’s initial speed? Answer the same question for a body sliding down a curved surface.Consider the following scenario. A car for which friction is not negligible accelerates from rest down a hill, running out of gasoline after a short distance (see below). The driver lets the car coast farther down the hill, then up and over a small crest. He then coasts down that hill into a gas station, where he brakes to a stop and fills that tank with gasoline. Identify the forms of energy the car has, and how they are changed and transferred in this series of events.A dropped ball bounces to one-half its original height. Discuss the energy transformations that take place.“ E=K+Uconstant is a special case of the work energy theorem.” Discuss this statement.In a common physics demonstration, a bowling ball is suspended from the ceiling by a rope. The professor pulls the ball away from its equilibrium position and holds it adjacent to his nose, as shown below. He releases the ball so that it swings directly away from him. Does he get struck by the ball on its return swing? What is he trying to show in this demonstration?A child jumps tip and down on a bed, reaching a higher height after each bounce. Explain how the child can increase his maximum gravitational potential energy with each bounce.Can a non-conservative force increase the mechanical energy of the system?Neglecting air resistance, how much would I have to raise the vertical height if I wanted to double the impact speed of a falling object?A box is dropped onto a spring at its equilibrium position. The spring compresses with the box attached and comes to rest. Since the spring is in the vertical position, does the change in the gravitational potential energy of the box while the spring is compressing need to be considered in this problem?Using values from Table 8.1, how many DNA molecules could be broken by the energy carried by a single electron in the beam of an old-fashioned TV tube? (These electrons were not dangerous in themselves, but they did create dangerous X-rays. Later-model tube TVs had shielding that absorbed X-rays before they escaped and exposed viewers.)If the energy in fusion bombs were used to supply the energy needs of the world, how many of the 9-megaton variety would be needed for a year’s supply of energy (using data from Equation 8.7)? U(x)=12kx2=const.A camera weighing 10 N falls from a small drone hovering 20 m overhead and enters free fall. What is the gravitational potential energy change of the camera from the drone to the ground if you take a reference point of (a) the ground being zero gravitational potential energy? (b) The drone being zero gravitational potential energy? What is the gravitational potential energy of the camera (c) before it falls from the drone and (d) after the camera lands on the ground if the reference point of zero gravitational potential energy is taken to be a second person looking out of a building 30 m from the ground?Someone drops a 50 — g pebble off of a docked cruise ship, 70.0 m from the water line. A person on a dock 3.0 m from the water line holds out a net to catch the pebble. (a) How much work is done on the pebble by gravity during the drop? (b) What is the change in the gravitational potential energy during the drop? If the gravitational potential energy is zero at the water line, what is the gravitational potential energy (c) when the pebble is dropped? (d) When it reaches the net? What if the gravitational potential energy was 30.0 Joules at water level? (e) Find the answers to the same questions in (c) and (d).A cat’s crinkle ball toy of mass 15 g is thrown straight up with an initial speed of 3 m/s. Assume in this problem that air drag is negligible. (a) What is the kinetic energy of the ball as it leaves the hand? (b) How much work is done by the gravitational force during the ball’s rise to its peak? (c) What is the change in the gravitational potential energy of the ball during the rise to its peak? (d) If the gravitational potential energy is taken to be zero at the point where it leaves your hand, what is the gravitational potential energy when it reaches the maximum height? (e) What if the gravitational potential energy is taken to be zero at the maximum height the ball reaches, what would the gravitational potential energy be when it leaves the hand? (f) What is the maximum height the ball reaches?A force F(x)=(3.0/x)N acts on a particle as it moves along the positive x-axis. (a) How much work does the force do on the particle as it moves from x=2.0 m to x=5.0 m? (b) Picking a convenient reference point of the potential energy to be zero at x=, find the potential energy for this force.A force F(x)=(5.0x2+7.0x)N acts on a particle as it moves along the positive x-axis. (a)How much work does the force do on the particle as it moves from x2.0 to x=5.0 m? (b) Picking a convenient reference point of the potential energy to be zero at x=, find the potential energy for this force.Find the force corresponding to the potential energy U(x)=a/x=b/x2.The potential energy function for either one of the two atoms in a diatomic molecule is often approximated by U(x)=a/x12b/x6 where x is the distance between the atoms. (a) At what distance of separation does the potential energy have a local minimum (x=) ? What is the force on an atom at this separation? (c) How does the force vary with the separation distance?A particle of mass 2.0 kg moves under the influence of the force F(x)=(3/x)N. If its speed at x=2.0 m is v=6.0 m/s, what is its speed at x = 7.0 m?A particle of mass 2.0 kg moves under the influence of the force F(x)=(5x2=7x)N. If its speed at x=4.0 m is v=20.0 m/s, what is its speed at x=4.0m ?A crate on rollers is being pushed without frictional loss of energy across the floor of a freight car (see the following figure). The car is moving to the right with a constant speed v0 . If the crate starts at rest relative to the freight car, then from the work-energy theorem, Fd=mv2/2, where d, the distance the crate moves, and v , the speed of the crate, are both measured relative to the freight car. (a) To an observer at rest beside the tracks, what distance d is the crate pushed when it moves the distance d in the car? (b) What are the crate’s initial and final speeds v0 and v as measured by the observer beside the tracks? (c) Show that Fd=m(v)2/2m(v0)2/2 and, consequently, that work is equal to the change in kinetic energy in both reference systems.A boy throws a ball of mass 0.25 kg straight upward with an initial speed of 20 m/s When the ball returns the boy, its speed is 17 m/s How much much work does air resistance do on the ball during its flight?A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance?Using energy considerations and assuming negligible air resistance, show that a rock thrown from a bridge 20.0 m above water with an initial speed of 15.0 m/s strikes the water with a speed of 24.8 m/s independent of the direction tin own. (Hint. show that (Ki+Ui=Kf+Uf)A 1.0-kg ball at the end of a 2.0-m string swings in a vertical plane. At its lowest point the ball is moving with a speed of 10 m/s. (a) What is its speed at the top of its path? (b) What is the tension in the string when the ball is at the bottom and at the top of its path?Ignoring details associated with friction, extra forces exerted by arm and leg muscles, and other factors, we can consider a pole vault as the conversion of an athlete’s running kinetic energy to gravitational potential energy. If an athlete is to lift his body 4.8 m during a vault, what speed must he have when he plants his pole?Tarzan grabs a vine hanging vertically from a tall tree when he is running at 9.0 m/s. (a) How high can he swing upward? (b) Does the length of the vine affect this height?Assume that the force of a bow on an arrow behaves like the spring force. In aiming the arrow, an archer pulls the bow back 50 cm and holds it in position with a force of 150 N. If the mass of the arrow is 50 g and the “spring” is massless, what is the speed of the arrow immediately after it leaves the bow?A 100 — kg man is skiing across level ground at a speed of 8.0 m/s when he comes to the small slope 1.8 m higher than ground level shown in the following figure. (a) If the skier coasts up the bill, what is his speed when he reaches the top plateau? Assume friction between the snow and skis is negligible. (b) What is his speed when he reaches the upper level if an 80 — N frictional force acts on the skis?A sled of mass 70 kg starts from rest and slides down a 10 incline 80 m long. It then travels for 20 m horizontally before starting back up an 8° incline. It travels 80 m along this incline before coming to rest. What is the magnitude of the net work done on the sled by friction?A girl on a skateboard (total mass of 40 kg) is moving at a speed of 10 m/s at the bottom of a long ramp. The ramp is inclined at 20 with respect to the horizontal. If she travels 14.2 m upward along the ramp before stopping, what is the net frictional force on her?A baseball of mass 0.25 kg is hit at home plate with a speed of 40 m/s. When it lands in a seat in the left-field bleachers a horizontal distance 120 m from home plate, it is moving at 30 m/s. If the ball lands 20 m above the spot where it was hit, how much work is done on it by air resistance?A small block of mass in slides without friction around the loop-the-loop apparatus shown below . (a) If the block starts from rest at A, what is its speed at B? (b) What is the force of the track on the block at B?The massless spring of a spring gun has a force constant k=12N/cm . When the gun is aimed vertically, a 15-g projectile is shot to a height of 5.0 m above the end of the expanded spring. (See below.) How much was the spring compressed initially?A small ball is tied to a string and set rotating with negligible friction in a vertical circle. Prove that the tension in the string at the bottom of the circle exceeds that at the top of the circle by eight times the weight of the ball. Assume the ball’s speed is zero as it sails over the top of the circle and there is no additional energy added to the ball during rotation.A mysterious constant force of 10 N acts horizontally on everything. The direction of the force is found to be always pointed toward a wall in a big hall. Find the potential energy of a particle due to this force when it is at a distance x from the wall, assuming the potential energy at the wall to be zero.A single force F(x)=4.0x (in newtons) acts on a 1.0-kg body. When x=3.5 m, the speed of the body is 4.0 m/s. What is its speed at x = 2.0 m?A particle of mass 4.0 kg is constrained to move along the x -axis under a single force F(x)=cx3 , where c=8.0N/m3 . The particle’s speed at A, where xA=1.0m , is 6.0 m/s. What is its speed at B, where xB=2.0m ?The force on a particle of mass 2.0 kg varies with position according to F(x)=3.0x2 (x in meters, F(x) in newtons). The particle’s velocity at x=2.0m is 5.0 m/s. Calculate the mechanical energy of the particle using (a) the origin as the reference point and (b) x = 4.0 m as the reference point. (c) Find the particle’s velocity at x=1.0m . Do this part of the problem for each reference point.A 4.0-kg particle moving along the x -axis is acted upon by the force whose functional form appears below. The velocity of the particle at x = 0 is v = 6.0 m/s. Find the particle’s speed at x=(a)2.0m, 2.0 (b)4.0 m. (c) 10.0m, (d) Does the particle turn around at some point and head back toward the origin? (e) Repeat part (d) if v = 2.0 m/s at x = 0.A particle of mass 0.50 kg moves along the x -axis with a potential energy whose dependence on x is shown below. (a) What is the force on the particle at x = 2.0, 5.0, 8.0, and 12 m? (b) If the total mechanical energy E of the particle is —6.0 J, what are the minimum and maximum positions of the particle? (c) What are these positions if E = 2.0 J? (d) If E = 16 J, what are the speeds of the particle at the positions listed in part (a)?(a) Sketch a graph of the potential energy function U(x)=kx2/2+Aex2 where k , A, and are constants. (b) What is the force corresponding to this potential energy? (c) Suppose a particle of mass in moving with this potential energy has a velocity v when its position is x = . Show that the particle does not pass 2+2 through the origin unless Amv2=k22(1e a 2 ) .In the cartoon movie Pocahontas (https://openstaxcollege.org/I/2lpocahontclip) Pocahontas runs to the edge of a cliff and jumps off, showcasing the fun side of her personality. (a) If she is running at 3.0 m/s before jumping off the cliff and she hits the water at the bottom of the cliff at 20.0 m/s, how high is the cliff? Assume negligible air drag in this cartoon. (b) If she jumped off the same cliff from a standstill, how fast would she be falling right before she hit the water?In the reality television show “Amazing Race” (https://openstaxcollege.org/l/2lamazraceclip), a contestant is firing 12-kg watermelons from a slingshot to hit targets down the field. The slingshot is pulled back 1.5 m and the watermelon is considered to be at ground level. The launch point is 0.3 m from the ground and the targets are 10 m horizontally away. Calculate the spring constant of the slingshot.In the Back to the Future movies (https://openstaxcollege.org/l/2lbactofutclip), a DeLorean car of mass 1230 kg travels at 88 miles per hour to venture back to the future. (a) What is the kinetic energy of the DeLorean? (b) What spring constant would be needed to stop this DeLorean in a distance of 0.1m?In the Hunger Games movie (https://openstaxcollege.org/l/21HungGamesclip), Katniss Everdeen fires a 0.0200-kg arrow from ground level to pierce an apple up on a stage. The spring constant of the bow is 330 N/m and she pulls the arrow back a distance of 0.55 m. The apple on the stage is 5.00 m higher than the launching point of the arrow. At what speed does the arrow (a) leave the bow? (b) strike the apple?In a “Top Fail” video (https://openstaxcollege.org/l/21topfailvideo), two women run at each other and collide by hitting exercise balls together. If each woman has a mass of 50 kg, which includes the exercise ball, and one woman runs to the right at 2.0 m/s and the other is running toward her at 1.0 m/s, (a) how much total kinetic energy is there in the system? (b) If energy is conserved after the collision and each exercise ball has a mass of 2.0 kg, how fast would the balls fly off toward the camera?In a Coyote/Road Runner cartoon clip (https://openstaxcollege.org/l/21coyroadcarcl), a spring expands quickly and sends the coyote into a rock. If the spring extended 5 m and sent the coyote of mass 20 kg to a speed of 15 m/s, (a) what is the spring constant of this spring? (b) If the coyote were sent vertically into the air with the energy given to him by the spring, how high could he go if there were no non-conservative forces?In an iconic movie scene, Forrest Gump (https://openstaxcollege.org/l/21ForrGumpvid) runs around the country. If he is running at a constant speed of 3 m/s, would it take him more or less energy to run uphill or downhill and why?In the movie Monty Python and the Holy Grail (https://openstaxcollege. org/l/21monpytmovcl) a cow is catapulted from the top of a castle wall over to the people down below. The gravitational potential energy is set to zero at ground level. The cow is launched from a spring of spring constant 1.1 104 N/m that is expanded 0.5 m from equilibrium. If the castle is 9.1 m tall and the mass of the cow is 110 kg, (a) what is the gravitational potential energy of the cow at the top of the castle? (b) What is the elastic spring energy of the cow before the catapult is released? (c) What is the speed of the cow right before it lands on the ground?A 60.0-kg skier with an initial speed of 12.0 m/s coasts up a 2.50-m high rise as shown. Find her final speed at the top, given that the coefficient of friction between her skis and the snow is 0.80.(a) How high a hill can a car coast up (engines disengaged) if work done by friction is negligible and its initial speed is 110 km/h? (b) if, in actuality, a 750-kg car with an initial speed of 110 km/h is observed to coast up a hill to a height 22.0 m above its starting point, how much thermal energy was generated by friction? (c) What is the average force of friction if the hill has a slope of 2.5 above the horizontal?A 5.00105kg subway train is brought to a stop from a speed of 0.500 m/s in 0.400 m by a large spring bumper at the end of its track. What is the spring constant k of the spring?A pogo stick has a spring with a spring constant of 2.5104N/m , which can be compressed 12.0 cm. To what maximum height from the uncompressed spring can a child jump on the stick using only the energy in the spring, if the child and stick have a total mass of 40 kg?A block of mass 500 g is attached to a spring of spring constant 80 N/m (see the following figure). The other end of the spring is attached to a support while the mass rests on a rough surface with a coefficient of friction of 0.20 that is inclined at angle of 300 . The block is pushed along the surface till the spring compresses by 10 cm and is then released from rest. (a) How much potential energy was stored in the block-spring-support system when the block was just released? (b) Determine the speed of the block when it crosses the point when the spring is neither compressed nor stretched. (c) Determine the position of the block where it just comes to rest on its way up the incline.A block of mass 200 g is attached at the end of a massless spring of spring constant 50 N/m. The other end of the spring is attached to the ceiling and the mass is released at a height considered to be where the gravitational potential energy is zero. (a) What is the net potential energy of the block at the instant the block is at the lowest point? (b) What is the net potential energy of the block at the midpoint of its descent? (c) What is the speed of the block at the midpoint of its descent?A T-shirt cannon launches a shirt at 5.00 m/s from a platform height of 3.00 m from ground level. How fast will the shirt be traveling if it is caught by someone whose hands are (a) 1.00 m from ground level? (b) 4.00 m from ground level? Neglect air drag.A child (32 kg) jumps up and down on a trampoline. The trampoline exerts a spring restoring force on the child with a constant of 5000 N/m. At the highest point of the bounce, the child is 1.0 m above the level surface of the trampoline. What is the compression distance of the trampoline? Neglect the bending of the legs or any transfer of energy of the child into the trampoline while jumping.Shown below is a box of mass m1 that sits on a frictionless incline at an angle above the horizontal =30. This box is connected by a relatively massless string, over a frictionless pulley, and finally connected to a box at rest over the ledge, labeled m2 . If m 1 and m2 are a height h above the ground and m2m1: (a) What is the initial gravitational potential energy of the system? (b) What is the final kinetic energy of the system?A massless spring with force constant k=200N/m hangs from the ceiling. A 2.0-kg block is attached to the free end of the spring and released. If the block falls 17 cm before starting back upwards, how much work is done by friction during its descent?A particle of mass 2.0 kg moves under the influence of the force F(x)=(5x2=7x)N . Suppose a frictional force also acts on the particle. If the particle’s speed when it starts at x=4.0 m is 0.0 m/s and when it arrives at x=4.0 m is 9.0 m/s, how much work is done on it by the frictional force between x=4.0 m and x=4.0 m?Block 2 shown below slides along a frictionless table as block 1 falls. Both blocks are attached by a frictionless pulley. Find the speed of the blocks after they have each moved 2.0 m. Assume that they start at rest and that the pulley has negligible mass. Use m1 = 2.0 kg and m2 = 4.0 kg.A body of mass m and negligible size starts from rest and slides down the surface of a frictionless solid sphere of radius R. (See below.) Prove that the body leaves the sphere when =cos1(2/3).A mysterious force acts on all particles along a particular line and always points towards a particular point P on the line. The magnitude of the force on a particle increases as the cube of the distance from that point; that is Fr3 , if the distance from P to the position of the particle is r. Let b be the proportionality constant, and write the magnitude of the force as F=br3. Find the potential energy of a particle subjected to this force when the particle is at a distance D from P, assuming the potential energy to be zero when the particle is at P.An object of mass 10 kg is released at point A, slides to the bottom of the 30 incline, then collides with a horizontal massless spring, compressing it a maximum distance of 0.75 m. (See below.) The spring constant is 500 M/m, the height of the incline is 2.0 m, and the horizontal surface is frictionless. (a) What is the speed of the object at the bottom of the incline? (b) What is the work of friction on the object while it is on the incline? (c) The spring recoils and sends the object back toward the incline. What is the speed of the object when it reaches the base of the incline? (d) What vertical distance does it move back up the incline?