Concept explainers
A 10-kg block is attached to spring A and connected to spring B by a cord and pulley. The block is held in the position shown with both springs unstretched when the support is removed and the block is released with no initial velocity. Knowing that the constant of each spring is 2 kN/m, determine (a) the velocity of the block after it has moved down 50 mm, (b) the maximum velocity achieved by the block.
Fig. P13.30
(a)
Find the velocity (v) of the block after it has moved down
Answer to Problem 13.30P
The velocity (v) of the block after it has moved down
Explanation of Solution
Given information:
The mass of the block (m) is
The spring constant at A
The spring constant at B
The depth where the spring A moves down
Assume the acceleration due to gravity (g) is
Calculation:
Show the free body diagram of the block with two spring’s attachment acting as in Figure (1).
Calculate the depth of spring B moves down due to block
Substitute
Calculate the weight of the block (W) using the relation:
Substitute
Here, the initial kinetic energy
Calculate the final kinetic energy
Substitute
Calculate the work done
Substitute
Calculate the work done
Substitute
Calculate the work done
Substitute
Calculate the total work done
Substitute
Use work and energy principle which states that kinetic energy of the particle at a displaced point can be obtained by adding the initial kinetic energy and the work done on the particle during its displacement.
Find the velocity (v) of the block after it has moved down
Substitute 0 for
Therefore, the velocity (v) of the block after it has moved down
(b)
Find the maximum velocity
Answer to Problem 13.30P
The maximum velocity
Explanation of Solution
Given information:
The mass of the block (m) is
The spring constant at A
The spring constant at B
The depth where the spring A moves down
Assume the acceleration due to gravity (g) is
Calculation:
Assume x be the distance moved down by the
Calculate the work done
Substitute
Calculate the work done
Substitute
Calculate the work done
Substitute
Calculate the total work done
Substitute
Differentiate the above equation with respect to ‘x’.
Substitute
Substitute
Use work and energy principle which states that kinetic energy of the particle at a displaced point can be obtained by adding the initial kinetic energy and the work done on the particle during its displacement.
Find the maximum velocity
Substitute 0 for
Therefore, the maximum velocity
Want to see more full solutions like this?
Chapter 13 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
- SHOW COMPLETE SOLUTION A spring is used to stop 50kg block moving down a 20 degrees inclined plane. The spring has a constant K= 30KN/M and is held by the cables so that it is initially compressed 50mm. Knowing that the velocity of the package is 2m/s when it is 8m from the spring and assuming the kinetic coefficient of friction between the package and the incline is 0.20. Determine: A. Final kinetic energy B. Maximum additional deformation of spring in bringing the block to rest C. Frictional forcearrow_forward13.30 A 10-kg block is attached to spring A and connected to spring B by a cord and pulley. The block is held in the position shown with both springs unstretched when the support is removed and the block is released with no initial velocity. Knowing that the constant of each spring is 2 kN/m, determine (a) the velocity of the block after it has moved down 50 mm, (b) the maximum velocity achieved by the block. k = 2 kN/m B wwwwwww 10 kg A C k = 2 kN/marrow_forward12.92 Two 2.6-lb collars A and B can slide without friction on a frame, con- sisting of the horizontal rod OE and the vertical rod CD, which is free to rotate about CD. The two collars are connected by a cord running over a pulley that is attached to the frame at O, and a stop prevents collar B from moving. The frame is rotating at the rate 0 = 12 rad/s and r = 0.6 ft when the stop is removed, allowing collar A to move out along rod OE. Neglecting friction and the mass of the frame, deter- mine, for the position r = 1.2 ft, (a) the transverse component of the velocity of collar A, (b) the tension in the cord and the acceleration of collar A relative to the rod OE. D B Fig. P12.92 A Earrow_forward
- A thin circular rod is supported in a vertical plane by a bracket at A. Attached to the bracket and loosely wound around the rod is a spring of constant k= 3 lb/ft and undeformed length equal to the arc of circle AB. An 8-oz collar C , not attached to the spring, can slide without friction along the rod. Knowing that the collar is released from rest at an angle 0 with the vertical, determine (a) the smallest value of 0 for which the collar will pass through D and reach point A, (b) the velocity of the collar as it reaches point A.arrow_forward4-12 PROBLEM 13.29 A 20-lb block is attached to spring A and connected to spring B by a cord and pulley. The block is held in the position shown with both springs unstretched when the support is removed and the block is released with no initial velocity. Knowing that the constant of each spring is 12 lb/in., determine (a) the velocity of the block after it has moved down 2 in., (b) the maximum velocity achieved by the block. v = 1.638 ft/s Vmax = 1.892 ft/sarrow_forwardQuestion 1. A spring is used to stop a 60 kg package which is sliding on a horizontal surface. The spring has a constant k = 20 kN/m and is held by cables so that it is initially compressed 120 mm. The package has a velocity of 2.5 m/s in the position shown and the maximum deflection of the spring is 40 mm. Determine (a) the coefficient of kinetic friction between the package and surface and (b) the velocity of the package as it passes again through the position shown. 2.5 m/s Cable 60 kg- 600 mm-arrow_forward
- A 32,000-lb airplane lands on an aircraft carrier and is caught by an arresting cable. The cable is inextensible and is paid out at A and b from mechanisms located below deck and consisting of pistons moving in long oil-filled cylinders. Knowing that the piston-cylinder system maintains a constant tension of 85 kips in the cable during the entire landing, determine the landing speed of the airplane if it travels a distance d= 95 ft after being caught by the cable.arrow_forward8 in. 6 in. B PROBLEM 13.58 A 3-lb collar is attached to a spring and slides without friction along a circular rod in a horizontal plane. The spring has an undeformed length of 7 in. and a constant k = 1.5 lb/in. Knowing that the collar is in equilibrium at A and is given a slight push to get it moving, determine the velocity of the collar (a) as it passes through B, (b) as it passes through C. VB = = 11.66 ft/s 15.01 ft/s VC =arrow_forwardww. E L d Vo В A bullet weighing 0.08 lb is fired with a horizontal velocity of 1800 ft/s into the lower end of a slender 15-lb bar of length L = 30 in. The slender bar is attached at E to an initially unstretched spring of constant k= 40 lb/ft as shown. Knowing that h=10 in and d= 25in. and that the bar is initially at rest. Using the impulse - Momentum principle, determine: %3D (a) the angular velocity of the bar immediately after the bullet becomes embedded. (b) the impulsive reaction at C Assuming that the bullet becomes embedded in 0.001s and the stretch distance of the spring is 2 in immediately after impact of the bullet.arrow_forward
- 9. A thin circular rod is supported in a vertical plane by a bracket at A. Attached to the bracket and loosely wound around the rod is a spring of constant k =3 lb/ft and undeformed length equal to the arc of circle AB. An 8-oz collar C, not attached to the spring, can slide without friction along the rod. Knowing that the collar is released from rest at an angle 0 with the vertical, determine (a) the smallest value of q for which the collar will pass through Dand reach Point A, (b) the velocity of the collar as it reaches Point A. |D 12 in. A | B oparrow_forwardA piston of mass m and cross-sectional area A is in equilibrium under the pressure p at the center of a cylinder closed at both ends. Assuming that the piston is moved to the left a distance a 2 and released, and knowing that the pressure on each side of the piston varies inversely with the volume, determine the velocity of the piston as it again reaches the center of the cylinder. Neglect friction between the piston and the cylinder and express your answer in terms of m, a, P and A.arrow_forwardSolve Prob. 13.68 assuming the kinetic coefficient of friction between the package and the incline is 0.2.Reference to Problem 13.68:A spring is used to stop a 50-kg package that is moving down a 20° incline. The spring has a constant k = 30 kN/m and is held by cables so that it is initially compressed 50 mm. Knowing that the velocity of the package is 2 m/s when it is 8 m from the spring and neglecting friction, determine the maximum additional deformation of the spring in bringing the package to rest.arrow_forward
- Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
- Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY