Concept explainers
Consider the simple pendulum shown in Fig. 6-7. (a) If it is released from point-A, what will be the speed of the ball as it passes through point-C? (b) What is the ball’s speed at point-B? [Hint: How far has it fallen upon arriving at point-B?]
Fig. 6-7
(a)
The speed of the ball at point C attached to a string of length
Answer to Problem 45SP
Solution:
Explanation of Solution
Given data:
Refer to Fig. 6-7.
The length of the string is
Formula used:
Understand that, for a simple pendulum at its lowest point, the kinetic energy is maximum and the potential energy is zero, while at its highest point, the potential energy is maximum and the kinetic energy is zero.
The expression for change in kinetic energy is written as:
Here,
The expression for change in potential energy is written as:
Here,
Explanation:
There is no work done on the ball. Also, the ball’s total energy remains constant, that is, it loses a part of its potential energy but gains an equal amount ofkinetic energy. Therefore,
Substitute
The velocity at the highest point is zero. Therefore,
Recall the expression for the work energy theorem.
Substitute
Conclusion:
The speed of the ball at point C is
(b)
The speed of the ball at point B attached to a string of length
Answer to Problem 45SP
Solution:
Explanation of Solution
Given data:
Refer to Fig. 6-7.
The length of the string is
Formula used:
Understand that, for a simple pendulum at its lowest point, the kinetic energy is maximum and the potential energy is zero, while at its highest point, the potential energy is maximum and the kinetic energy is zero.
The expression for change in kinetic energy is written as:
Here,
The expression for change in potential energy is written as:
Here,
Explanation:
Draw the diagram of a simple pendulum.
There is no work done on the ball. Also, the ball’s total energy remains constant, that is, it loses a part of its kinetic energy but gains an equal amount of potential energy. Therefore,
Substitute
Consider the diagram and determine the distance between points
From previous part, the velocity at point C is
Recall the expression for the work energy theorem.
Substitute
Conclusion:
The speed of ball at point B is
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