
Concept explainers
(a)
The speed of the ball as it begins rolling without slipping.
(a)

Answer to Problem 107P
Explanation of Solution
Given:
Radius of the ball is
Initial speed of the ball is
The coefficient of kinetic friction between the ball and billiard table is
Forward spin of the ball just after its release is
Formula Used:
FIGURE: 1
Constant acceleration equation that relates the speed of the ball to the acceleration and time,
Where,
Referring to the force diagram shown in figure 1, applying Newton’s second law to the ball when it is rolling without slipping,
And
Where,
Calculation:
Where,
From equation
Substituting this in equation
Now,substituting the expression for
Substituting for
From equation
Moment of inertia with respect to an axis through the center of the ball is
Substituting for
Now let us write constant-acceleration equation that connects angular speed of the ball to the angular acceleration and time,
Imposing the condition for rolling the ball without slipping,
Substituting for
Now equate the equations
Substituting this
Conclusion:
The speed of the ball as it begins rolling without slipping is
(b)
The time the ball moves before it begins to rolling without slipping .
(b)

Answer to Problem 107P
Explanation of Solution
Given:
Radius of the ball is
Initial speed of the ball is
The coefficient of kinetic friction between the ball and billiard table is
Forward spin of the ball just after its release is
Formula Used:
FIGURE: 2
Constant acceleration equation that relates the speed of the ball to the acceleration and time,
Where,
Referring to the force diagram shown in figure 2, applying Newton’s second law to the ball when it is rolling without slipping,
And
Where,
Calculation:
Where,
From equation
Substituting this in equation
Now,substituting the expression for
Substituting for
From equation
Moment of inertia with respect to an axis through the center of the ball is
Substituting for
Now let us write constant-acceleration equation that connects angular speed of the ball to the angular acceleration and time,
Imposing the condition for rolling the ball without slipping,
Substituting for
Now equate the equations
Conclusion:
The time the ball moves before it begins to rolling without slipping
(c)
The distance slide down the lane by the ball before it begins rolling without slipping.
(c)

Answer to Problem 107P
Explanation of Solution
Given:
Radius of the ball is
Initial speed of the ball is
The coefficient of kinetic friction between the ball and billiard table is
Forward spin of the ball just after its release is
Calculation:
Let
Now let us write expression that relates
Average speed of the ball is,
Substituting this average speed in equation
Substituting for
Conclusion:
The distance slide down the lane by the ball before it begins rolling without slipping is
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