A ball of mass M = 5.00 kg and radius r = 5.00 cm is attached to one end of a thin, cylindrical rod of length L = 15.0 cm and mass m = 0.600 kg. The ball and rod, initially at rest in a vertical position and free to rotate around the axis shown in Figure P13.70, are nudged into motion. a. What is the rotational kinetic energy of the system when the ball and rod reach a horizontal position? b. What is the angular speed of the ball and rod when they reach a horizontal position? c. What is the linear speed of the center of mass of the ball when the ball and rod reach a horizontal position? d. What is the ratio of the speed found in part (c) to the speed of a ball that falls freely through the same distance?
FIGURE P13.70
(a)
The rotational kinetic energy of the system when the ball and rod reach a horizontal position.
Answer to Problem 70PQ
The rotational kinetic energy of the system when the ball and rod reach a horizontal position is
Explanation of Solution
For the isolated rod-ball-Earth system with no friction, the mechanical energy is conserved.
Here,
The initially the system has only potential energy and has no kinetic energy. At the final position, the entire potential energy is converted to kinetic energy. Thus, equation (I) can be modified as,
Write the expression for the initial potential energy of the given system.
Here,
Use equation (III) in (II) and solve for
For the uniform rod, the center of mass is at the mid-point. Since the length of the rod is
Conclusion:
Substitute
Therefore, the rotational kinetic energy of the system when the ball and rod reach a horizontal position is
(b)
The angular speed of the system when it reach the horizontal position.
Answer to Problem 70PQ
The angular speed of the system when it reach the horizontal position is
Explanation of Solution
The system can be assumed as, the sphere as a point particle which occupies at a distance
Write the expression for the rotational inertia of the given rod-ball system.
Here,
Write the expression for the rotational kinetic energy of the system at the horizontal position..
Here,
Solve equation (VI) for
Conclusion:
Substitute
Substitute
Therefore, the angular speed of the system when it reach the horizontal position is
(c)
The linear speed of the center of mass of the ball when the system reach the horizontal position.
Answer to Problem 70PQ
The linear speed of the center of mass of the ball when the system reach the horizontal position is
Explanation of Solution
It is obtained that the angular speed of the system when it reach the horizontal position is
Write the expression for the linear speed in terms of the angular speed.
Here,
Conclusion:
Substitute
Therefore, the linear speed of the center of mass of the ball when the system reach the horizontal position is
(d)
The ratio of the speed of the ball when the system is at the horizontal position and the speed of the ball that falls freely through the same distance.
Answer to Problem 70PQ
The ratio of the speed of the ball when the system is at the horizontal position and the speed of the ball that falls freely through the same distance is
Explanation of Solution
It is obtained that the linear speed of the center of mass of the ball when the system reach the horizontal position is
Write the expression for the speed of the freely balling ball.
Here,
Since the ball falls from rest, and reaches the ground, the initial speed is zero, and the final height is zero. Thus, equation (IX) can be modified and solved for
Conclusion:
Substitute
This indicates that the speed of the ball when the system is at the horizontal position and the speed of the ball that falls freely through the same distance are the same so that their ratio is obtained as,
Therefore, the ratio of the speed of the ball when the system is at the horizontal position and the speed of the ball that falls freely through the same distance is
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Chapter 13 Solutions
Physics for Scientists and Engineers: Foundations and Connections
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