Concept explainers
(a)
The speed of the ball just after impact.
(a)
Answer to Problem 104P
Explanation of Solution
Given:
Mass of a uniform solid ball =
Radius of the ball =
The height above the horizontal surface at which the force is applied on the ball,
During the impact, the force (F) increases from
Therefore, average force,
Formula used:
Applying impulse-momentum theorem to the ball,
Where
Calculation:
FIGURE:
Substituting numerical values in equation
Conclusion:
The speed of the ball just after impact is
(b)
The angular speed of the ball after impact.
(b)
Answer to Problem 104P
Explanation of Solution
Given:
Mass of a uniform solid ball =
Radius of the ball =
The height above the horizontal surface at which the force is applied on the ball,
During the impact, the force (F) increases from
Therefore, average force,
Formula used:
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
From equation
Substituting the expression for
Calculation:
FIGURE: 2
Substituting the numerical values in equation
Conclusion:
The angular speed of the ball after impact is
(c)
The speed of the ball when it begins to roll without slipping.
(c)
Answer to Problem 104P
Explanation of Solution
Given:
Mass of a uniform solid ball =
Radius of the ball =
The height above the horizontal surface at which the force is applied on the ball,
During the impact, the force (F) increases from
Therefore, average force,
Coefficient of kinetic friction,
Formula used:
Constant acceleration equation that relates the speed of the ball to the acceleration and time,
Where,
Referring to the force diagram shown in figure 3, applying Newton’s second law to the ball,
And
Where,
But,
Where,
From equation
Substituting this in equation
Substituting the expression for
Substituting
From equation
Substituting for
Now let us write constant-acceleration equation that connects angular speed of the ball to the angular acceleration and time,
When the ball rolls without slipping
From equation
Hence,
Now equating the expressions
On rearranging,
Calculation:
FIGURE:3
Substituting the numerical values in equation
Substituting the numerical values in equation
Conclusion:
The speed of the ball when it begins to roll without slipping is
(d)
The distance travelled by the ball along the surface before it begins to roll without slipping.
(d)
Answer to Problem 104P
Explanation of Solution
Given: Coefficient of kinetic friction,
Formula used:
The distance travelled by the ball in time
Since,
Where,
Calculation:
FIGURE: 4
From the part
Substituting the numerical values in equation
Conclusion:
The distance travelled by the ball along the surface before it begins to roll without slipping is
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Chapter 9 Solutions
Physics for Scientists and Engineers, Vol. 3
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