Concept explainers
(a)
The angular speed of the ball just after the blow.
(a)
Answer to Problem 106P
Explanation of Solution
Given: The horizontal force applied on a billiard ball at a distance
Speed of the ball just after the blow is
The coefficient of kinetic friction between the ball and billiard table is
Formula Used:
Newton’s second law of motion in rotational form
Calculation:
FIGURE: 1
Applying Newton’s second law in rotational form to ball,
Where,
Moment of inertia with respect to an axis through the center of mass of the ball is
Substituting this in equation
Applying impulse-momentum theorem to the ball,
Where,
From equation
Substituting the expression for
Substituting
Conclusion:
The angular speed of the ball just after the blow is
(b)
The speed of the ball once it begins to roll without slipping.
(b)
Answer to Problem 106P
Explanation of Solution
Given: Speed of the ball just after the blow is
The coefficient of kinetic friction between the ball and billiard table is
Formula Used:
FIGURE: 2
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,
But,
Where,
Calculations:
From equation
Substituting this in equation
From equation
Moment of inertia with respect to an axis through the center of mass 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,
Now,substituting the expression for
Constant acceleration equation that relates the speed of the ball to the acceleration and time,
Where,
Substituting for
Imposing the condition for rolling the ball without slipping,
Substituting this
Conclusion:
The speed of the ball once it begins to roll without slipping is
(c)
The kinetic energy of the ball just after the hit.
(c)
Answer to Problem 106P
Explanation of Solution
Given: The horizontal force applied on a billiard ball at a distance
Speed of the ball just after the blow is
The coefficient of kinetic friction between the ball and billiard table is
Formula Used:
Initial kinetic energy of the ball can be written as,
Where,
Substituting the expressions for
Where,
Moment of inertia,
Substituting these in equation
Conclusion:
The kinetic energy of the ball just after the hit is
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Chapter 9 Solutions
Physics for Scientists and Engineers
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