Concept explainers
In a game of pool, ball A is rolling without slipping with a velocity
Fig. P17.133
a)
The linear and angular velocity of each ball immediately after the impact.
Answer to Problem 17.133P
The linear velocity of ball A immediately after the impact is
The angular velocity of ball A immediately after the impact is
The linear velocity of ball B immediately after the impact is
The angular velocity of ball B immediately after the impact is
Explanation of Solution
Given information:
The mass of the each ball is m.
The radius of the each ball is r.
The velocity of the ball A before the impact is
The coefficient of kinetic friction between a ball and the table
Calculation:
Write the equation of moment of inertia
Analyze the impact of ball A.
Here, G is the mass center of ball A.
Consider the conservation of momentum principle.
Sketch the impulse and momentum diagram of the ball A as shown in Figure 1.
Here,
Refer Figure (1).
Consider the kinematics in position 1.
Write the equation of angular velocity
Consider the horizontal components of forces.
Consider the vertical components of forces.
Take moments about y axis.
Take moments about x axis.
Analyze the impact of ball B.
Here, G is the mass center of ball B.
Consider the conservation of momentum principle.
Sketch the impulse and momentum diagram of the ball B as shown in Figure (2).
Here,
Refer Figure (2),
Consider the horizontal components of forces.
Consider the vertical components of forces.
Take moments about y axis.
Take moments about x axis.
Add Equations (1) and (5).
The impact is perfectly plastic. Therefore ,the coefficient of restitution (e) is 1.
Consider the condition of impact equation.
Substitute 1 for e.
Find the horizontal components of linear velocity
Solve Equations (9) and (10) simultaneously.
Add Equations (9) and (10).
Find the horizontal components linear velocity
Substitute
Find the vertical components of linear velocity
Find the vertical components of linear velocity
Find the linear velocity
Substitute 0 for
Thus, the linear velocity of ball A immediately after the impact is
Find the linear velocity
Substitute
Thus, the linear velocity of ball B immediately after the impact is
Find the initial angular velocity of ball A using kinematics.
Find the angular velocity
Solve Equations (3) and (4) simultaneously.
Add Equations (3) and (4).
Substitute
Thus, the angular velocity of ball A immediately after the impact is
Find the angular velocity
From Equations (7) and (8).
Thus, the angular velocity of ball B immediately after the impact is
b)
Find the velocity of ball B after it has started rolling uniformly.
Answer to Problem 17.133P
The velocity of ball B after it has started rolling uniformly is
Explanation of Solution
Calculation:
Consider the motion after impact of ball B.
Consider C is the mass center of ball A.
Consider the conservation of momentum principle.
Sketch the motion of impulse and momentum diagram of the ball A after the impact as shown in Figure (3).
Here,
Consider the condition of rolling without slipping in kinematics.
Refer Figure (3).
Take moments about C:
Substitute
Find the velocity of sphere A after it has started rolling uniformly using the equation:
Substitute
Thus, the velocity of ball A after it has started rolling uniformly is
Consider the motion after impact of ball A.
Consider C is the mass center of ball A.
Consider the conservation of momentum principle.
Sketch the motion of impulse and momentum diagram of the ball B after the impact as shown in Figure (3).
Here, N is the normal force on ball A, F is the friction force between ball and floor, t is the rolling time,
Consider the condition of rolling without slipping in kinematics.
Refer Figure (3),
Take moments about C:
Substitute
Find the velocity of sphere B after it has started rolling uniformly using the equation:
Substitute
Thus, the velocity of sphere B after it has started rolling uniformly is
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