Review. An object with a mass of m = 5.10 kg is attached to the free end of a light string wrapped around a reel of radius R = 0.250 m and mass M = 3.00 kg. The reel is a solid disk, free to rotate in a vertical plane about the horizontal axis passing through its center as shown in Figure P10.45. The suspended object is released from rest 6.00 m above the floor. Determine
- (a) the tension in the string,
- (b) the acceleration of the object, and
- (c) the speed with which the object hits the floor.
- (d) Verify your answer to part (c) by using the isolated system (energy) model.
Figure P10.45
(a)
The tension acting on the string.
Answer to Problem 45P
The tension acting on the string is
Explanation of Solution
Consider the figure given below.
The forces acting on the reel are given in the figure 1. The gravitational force acting on the reel is
Write the expression for the gravitational force.
Here,
The reel can be considered as a uniform disk, let
Write the expression for the moment of inertia of the reel.
Here,
Write the expression for torque about the axis of rotation.
Here,
Equate
Here,
Substitute,
Write the expression of force in downward direction.
Conclusion:
Substitute,
Substitute,
Substitute,
Substitute,
Substitute, equation (VII) in equation (VIII) to obtain the value of acceleration.
Substitute,
Therefore, the tension acting on the string is
(b)
The acceleration of the object.
Answer to Problem 45P
The acceleration of the object is
Explanation of Solution
The acceleration is already calculated in part (a).
Conclusion:
Therefore, the acceleration of the object is
(c)
The speed with which object hits the ground.
Answer to Problem 45P
The speed with which object hits the ground is
Explanation of Solution
Write the Newton’s equation of motion for the final velocity.
Here,
Conclusion:
Substitute,
Therefore, the speed with which object hits the ground is
(d)
To verify the value of speed of the object using isolated system model.
Answer to Problem 45P
The value of acceleration obtained using both methods are same.
Explanation of Solution
Write the equation for the conservation energy in case of the given system.
Here,
Substitute,
Substitute,
Conclusion:
Substitute,
Therefore, the value of acceleration obtained using both methods are same.
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Chapter 10 Solutions
Principles of Physics: A Calculus-Based Text
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