Concept explainers
A tiny solid ball
Fig. 10-12
(a)
The speed of the ball as it passes through the point B as shown in Fig. 10-12.
Answer to Problem 55SP
Solution:
Explanation of Solution
Given data:
The moment of inertia of the solid ball is
The ball is released at point A. Therefore, the value of the initial velocity at that point is zero.
The movement of ball is as shown in Fig. 10-12;itfollows a circular path of radius
Formula used:
Write the expression for calculating thepotential energy of an object:
Here,
Write the expression for the total kinetic energy of a rolling object:
Here,
Write the expression for total energy of system:
Here,
Write the expression for conservation of energy:
Here,
Write the expression for linear velocity in terms of angular velocity:
Here,
Explanation:
Draw the arrangement of the system shown in Fig 1:
Consider the point B be the reference point, that is, the height of the ball is measured with respect to this point.
The expression forpotential energy of the ball at point A is
Here,
For point A, substitute
The expression forpotential energy of ball at point B is
Here,
For point B, substitute
The expression for the total kinetic energy of theball at point A is
Here,
Since the ball is released from rest at point A, its initial velocity is zero.
For point A, substitute
The expression for linear velocity in terms of angular velocity at point B is
Here,
At point B, substitute
Solve for
The expression for the total kinetic energy of ball at point B is
Here,
For point B, substitute
The expression for total energy of the system at point A is
Here,
At point A, substitute
The expression for the total energy of the system at point B is
Here,
At point B, substitute
The expression for conservation of energy for the point of observation A and point B is
Substitute
Solve for
Take square root both sides:
Conclusion:
The speed of the ball as it passes through the point B is
(b)
The speed of the ball as it passes through the point Cas shown in Fig. 10-12.
Answer to Problem 55SP
Solution:
Explanation of Solution
Given data:
The moment of inertia of the solid ball is
The ball is released at point A that is the velocity of the ball at this point is zero.
The movement of ball is shown in Fig. 10-12;it follows a circular path of radius
Formula used:
Write the expression for calculating the potential energy of an object:
Here,
Write the expression for total kinetic energy of a rolling object:
Here,
Write the expression for total energy of a system:
Here,
Write the expression for conservation of energy:
Here,
Write the expression for linear velocity in terms of angular velocity:
Here,
Explanation:
Let point B be the reference point, that is, the height of the ball is measured with respect to this point.
Recall the total energy at point A from subpart (a):
The expression forpotential energy of ball at point C is
Here,
The expression for height of point C is
Here,
Substitute
For point C, substitute
The expression for total kinetic energy of a rolling object at point C is
Here,
The expression for linear velocity in terms of angular velocity is
Here,
At point C, substitute
Solve for
For point C, substitute
The expression for total energy of the system at point C is
Here,
At point C, substitute
The expression for conservation of energy for the point of observation A and C is
Substitute
Solve for
Further, solve for
Conclusion:
The speed of the ball as it passes through the point C is
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Chapter 10 Solutions
Schaum's Outline of College Physics, Twelfth Edition (Schaum's Outlines)
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