Concept explainers
(a)
To Show:
Path of the particle is circle of radius R.
(a)
Explanation of Solution
Given:
Mass of a particle
Position of a particle is given by
Radius of circle
Calculation:
To show that particle is moving in a circle take a dot product of
It is the equation of the circle with radius R and center at the origin. It is given, R
Conclusion:
It is shown that the path of this particle is the circle of radius R and center at the origin.
(b)
To Compute:
The velocity vectors.
To Show
:
(b)
Explanation of Solution
Given:
Mass of a particle
Position of a particle is given by
Radius of circle
Calculation:
Hence,
Conclusion:
It is shown that
(c)
To compute:
The acceleration vector
To Show:
The acceleration is directed towards the origin and has magnitude
(c)
Explanation of Solution
Given:
Calculation:
Hence,acceleration is directed towards the center.
Magnitude of acceleration is
But
Conclusion:
It is shown that the acceleration vector is directed towards the origin and has magnitude
(d)
To Find:
The magnitude and direction of net force acting on the particle.
(d)
Answer to Problem 130P
Magnitude of net force acting on the particle is
Explanation of Solution
Given:
Mass of a particle
Radius of circle
Formula Used:
Calculation:
The only force acting on the particle in a circular motion is the centripetal force. Hence the magnitude of the total force on the particle is
The direction of force is equal to the direction of centripetal acceleration i.e. towards the center, along the radius.
Conclusion:
The magnitude of the net force acting on the particle is
Want to see more full solutions like this?
Chapter 5 Solutions
Physics for Scientists and Engineers
- Using Figure 13.9, carefull sketch a free body diagram for the case of a simple pendulum hanging at latitude lambda, labeling all forces acting on the point mass,m. Set up the equations of motion for equilibrium, setting one coordinate in the direction of the centripetal accleration (toward P in the diagram), the other perpendicular to that. Show that the deflection angle , defined as the angle between the pendulum string and the radial direction toward the center of Earth, is given by the expression below. What is the deflection angle at latitude 45 degrees? Assume that Earth is a perfect sphere. tan(+)=gg2REtan , where is the angular velocity of Earth.arrow_forwardConsider a pulsar, a collapsed star of extremely high density, with a mass M equal to that of the Sun (1.98 * 10^30 kg), a radius R of only 12 km, and a rotational period T of 0.041 s. By what percentage does the free-fall acceleration g differ from the gravitational acceleration ag at the equator of this spherical star?arrow_forwardVery far from earth (at R=∞), a spacecraft has run out of fuel and its kinetic energy is zero. If only the gravitational force of the earth were to act on it (i.e., neglect the forces from the sun and other solar system objects), the spacecraft would eventually crash into the earth. The mass of the earth is Me and its radius is Re. Neglect air resistance throughout this problem, since the spacecraft is primarily moving through the near vacuum of space. Find the speed se of the spacecraft when it crashes into the earth. Express the speed in terms of Me, Re, and the universal gravitational constant G. Now find the spacecraft's speed when its distance from the center of the earth is R=αRe, where the coefficient α≥1. Express the speed in terms of se and α.arrow_forward
- Very far from earth (at R=∞), a spacecraft has run out of fuel and its kinetic energy is zero. If only the gravitational force of the earth were to act on it (i.e., neglect the forces from the sun and other solar system objects), the spacecraft would eventually crash into the earth. The mass of the earth is Me and its radius is Re. Neglect air resistance throughout this problem, since the spacecraft is primarily moving through the near vacuum of space. Find the speed se of the spacecraft when it crashes into the earth. Express the speed in terms of Me, Re, and the universal gravitational constant G. Use a conservation-law approach. Specifically, consider the mechanical energy of the spacecraft when it is (a) very far from the earth and (b) at the surface of the earth.arrow_forwardShow that the M.I. of an elliptic area of mass M and equation ax² + 2hxy + by² + 2gx + 2fy + c = 0, about a diameter parallel to the axis of x is a h g B h b f 8 f c M as 4 (ab-h²)² where A =arrow_forwardA tunnel is drilled from the surface of a planet straight down to its antipodal pointon the other side of the planet. A particle at the surface is dropped from rest intothe tunnel. Denote the position of the particle from the planet’s center at time t byr(t). Assume that the gravitational force the particle experiences is due only to themass of the planet within the sphere of radius r(t). Calculate the time (in seconds)the particle will take to reach the other end of the planet. Let the radius of the entireplanet be R = 20 × 106 m and take 4.9 as the value of garrow_forward
- Consider a pulsar, a collapsed star of extremely high density, with a mass M equal to that of the Sun (1.98 × 1030 kg), a radius R of only 17.4 km, and a rotational period T of 0.0448 s. By what percentage does the free-fall acceleration g differ from the gravitational acceleration ag at the equator of this spherical star?arrow_forwardAssume a planet is a uniform sphere of radius R that(somehow) has a narrow radial tunnel through its center. Also assume we can position an apple anywherealong the tunnel or outside the sphere. Let FR be the magnitudeof the gravitational force on the apple when it is located atthe planet’s surface. How farfrom the surface is there a pointwhere the magnitude is FR ifwe move the apple (a) awayfrom the planet and (b) intothe tunnel?arrow_forwardConsider a thin disc of radius R and consisting of a material with constant mass density (per unit of area) g. Use cylindrical coordinates, with the z-axis perpendicular to the plane of the disc, and the origin at the disc's centre. We are going to calculate the gravitational potential, and the gravitational field, in points on the z-axis only. the gravitational potential p(2) set up by that disc is given by dr'; ()² + z² sp(2) = 27Ggarrow_forward
- Calculate the parallelogram with sides: A (-6,4,-12, and B-(9,-6.18). The tuo vertirs are parallelarrow_forwardAn artificial satellite is launched from -6000™ 12742 *3000* point B on the equator by its carrier rocket and inserted into an elliptical orbit with perigee altitude of 3000 km. If the apogee altitude is to be 6000 km, compute a) The necessary perigee velocity vp and the corresponding apogee velocity v4. b) The velocity at point C where the altitude of VA satellite is 3500 km. 3500 c) The period t for a complete orbit. 2)arrow_forward(COLLAB Gauss's Law for Mass) Journey through the Center of the Earth. A 1024.kg blue ball is dropped from an initial z- position of 2 x 106 m through the center of a planet with radius 8.6 x 106 m. If the mass of the planet is 38.1 x 1015 kg, measure the displacement of the ball at timet = 9 s?arrow_forward
- Classical Dynamics of Particles and SystemsPhysicsISBN:9780534408961Author:Stephen T. Thornton, Jerry B. MarionPublisher:Cengage LearningPrinciples of Physics: A Calculus-Based TextPhysicsISBN:9781133104261Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningUniversity Physics Volume 1PhysicsISBN:9781938168277Author:William Moebs, Samuel J. Ling, Jeff SannyPublisher:OpenStax - Rice University
- Modern PhysicsPhysicsISBN:9781111794378Author:Raymond A. Serway, Clement J. Moses, Curt A. MoyerPublisher:Cengage Learning