A 10-lb collar is attached to a spring and slides without friction along a fixed rod in a vertical plane. The spring has an undeformed length of 14 in. and a constant k = 4 lb/in. Knowing that the collar is released from rest in the position shown, determine the force exerted by the rod on the collar at (a) point A, (b) point B. Both these points are on the curved portion of the rod.
Fig. P13.73
(a)
Find the force exerted by the rod on the collar at A
Answer to Problem 13.73P
The force exerted by the rod on the collar at A
Explanation of Solution
Given information:
The weight of the collar (W) is
The un-deformed length
The spring constant (k) is
The length of top support to point A
The length of point A to point B
The horizontal distance from weight to point A
The horizontal distance from point A to point B
The acceleration due to gravity (g) is
Calculation:
Calculate the mass of the collar (m) using the relation:
Substitute
Consider the position 1.
Calculate the length from weight to point B
Substitute
Calculate the stretch in rod
Substitute
Here, the kinetic energy at position 1
Calculate the potential energy in the position 1 due to elongation of the rod
Substitute
Here, the potential energy in the position 1 due to gravitation of the rod
Calculate the total potential energy
Substitute
Consider the position A.
Calculate the length at point A
Substitute
Calculate the stretch in rod
Substitute
Calculate the kinetic energy at position A
Here,
Substitute
Calculate the potential energy in the position A due to elongation of the rod
Substitute
Here, the potential energy in the position 1 due to gravitation of the rod
Calculate the total potential energy
Substitute
The expression for principle for conservation of energy as follows;
Substitute 0 for
Show the free body diagram of the point A with the forces acting as in Figure (1).
Calculate the normal acceleration at position A
Substitute
Calculate the spring force at position A
Substitute
Calculate the angle
Substitute
Calculate the force exerted by the rod on the collar in the point A
Substitute
Therefore, the force exerted by the rod on the collar at A
(b)
Find the force exerted by the rod on the collar at B
Answer to Problem 13.73P
The force exerted by the rod on the collar at B
Explanation of Solution
Given information:
The weight of the collar (W) is
The un-deformed length
The spring constant (k) is
The length of top support to point A
The length of point A to point B
The horizontal distance from weight to point A
The horizontal distance from point A to point B
The acceleration due to gravity (g) is
Calculation:
Consider the position B.
Calculate the length at point B
Substitute
Calculate the stretch in rod
Substitute
Calculate the kinetic energy at position B
Here,
Substitute
Calculate the potential energy in the position B due to elongation of the rod
Substitute
Calculate the potential energy in the position B due to gravitation of the rod
Substitute
Calculate the total potential energy
Substitute
The expression for principle for conservation of energy as follows;
Substitute 0 for
Show the free body diagram of the point B with the forces acting as in Figure (2).
Calculate the normal acceleration at position b
Substitute
Calculate the spring force at position B
Substitute
Calculate the force exerted by the rod on the collar in the point B
Substitute
Therefore, the force exerted by the rod on the collar at B
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Chapter 13 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
- A 10-lb collar is attached to a spring and slides without friction along a fixed rod in a vertical plane. The spring has an undeformed length of 14 in. and a constant k= 4 lb/in. Knowing that the collar is released from rest in the position shown, determine the force exerted by the rod on the collar at (a) point A, (b) point B. Both these points are on the curved portion of the rod.arrow_forward4. Rod OA rotates about O in a horizontal plane. The motion of the 0.5-lb collar B is defined by the relations r = 10 + 6 coSn t and e = 1 (4t? – 8t), where r is expressed in inches, t in seconds, and 0 in radians. Determine the radial and transverse components of the force exerted on the collar when (a) t= 0, (b) t = 0.5 s. Barrow_forwardThe double pulley shown has a weight of 35.0 lb and a centroidal radius of gyration of 5.0 in. Cylinder A (25.0 lb) and block B (16 lb) are attached to cords that wrap around pulleys in the manner shown. The coefficient of kinetic friction between block B and the surface is 0.25. Knowing that the system is released from rest at the position shown (h = 4 ft), determine the velocity of cylinder A when it strikes the ground. 6 in. A h 10 in. Barrow_forward
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- Activity 3. A body that weighs W Newtons falls from rest from a height of 600mm and strikes a spring whose scale is 7.00 N/mm. If the maximum compression of the spring is 150 mm, what is the value of W? Disregard the mass of the spring. (Ans. W=105 N) 600mm 150mmarrow_forwardTwo steel balls, each of mass m = 1.84 kg, are welded to a light rod of length L = 545 mm and negligible mass and are initially at rest on a smooth horizontal surface. The distance b = 111 mm. A horizontal force of magnitude F = 27 N is suddenly applied to the rod as shown. Determine (a) the magnitude of the instantaneous acceleration a of the mass center G and (b) the magnitude of the corresponding rate 0 at which the angular velocity of the assembly about G is changing with time. m Answers: a = 0= IN P 77 M m/s² rad/s²arrow_forwardTwo steel balls, each of mass m = 1.84 kg, are welded to a light rod of length L = 545 mm and negligible mass and are initially at rest on a smooth horizontal surface. The distance b= 111 mm. A horizontal force of magnitude F= 27 N is suddenly applied to the rod as shown. Determine (a) the magnitude of the instantaneous acceleration a of the mass center G and (b) the magnitude of the corresponding rate 0 at which the angular velocity of the assembly about G is changing with time. m Answers: a = O = IN 7. 77 M m/s² rad/s²arrow_forward
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