Disk B is at rest when it is brought into contact with disk A, which has an initial angular velocity ω0. (a) Show that the final angular velocities of the disks are independent of the coefficient of friction μk between the disks as long as μk ≠ 0. (b) Express the final angular velocity of disk A in terms of ω0 and the ratio of the masses of the two disks, mA/mB.
Fig. P16.43 and P16.44
(a)
Show that the final velocities of disk A and B are independent of coefficient of kinetic friction
Explanation of Solution
The mass of the disk A is
The mass of the disk B is
The initial angular velocity of the disk A is
The coefficient of the kinetic friction is
The radius of the disk A is
The radius of the disk B is
The acceleration due to gravity is g.
The time required for the disk to come to rest is t.
Calculation:
Calculate the mass moment of inertia of the disk A
Calculate the mass moment of inertia of the disk B
Calculate the load of the disk A
Calculate the load of the disk B
Show the free body diagram of the disk B as in Figure 1.
Here,
Refer to Figure 1.
Calculate the vertical forces by applying the equation of equilibrium:
Sum of vertical forces is equal to 0.
Substitute
Calculate the magnitude of the friction force
Substitute
Calculate the horizontal forces by applying the equation of equilibrium:
Sum of horizontal forces is equal to 0.
Substitute
Calculate the angular acceleration of the disk B
Calculate the moment about point B by applying the equation of equilibrium:
Substitute
Show the free body diagram of the disk A as in Figure 2.
Here,
Refer to Figure 2.
Calculate the horizontal forces by applying the equation of equilibrium:
Sum of horizontal forces is equal to 0.
Substitute
Calculate the vertical forces by applying the equation of equilibrium:
Sum of vertical forces is equal to 0.
Substitute
Calculate the angular acceleration of the disk A
Calculate the moment about point A by applying the equation of equilibrium:
Substitute
The angular velocity of the disk A
Substitute
The angular velocity of the disk B
Substitute
While there is no slipping between disk A and B, their velocity ratio is same.
Show the free body diagram of the system as in Figure 3.
Refer to Figure 3.
The velocity of pinion
Substitute
Calculate the angular velocity of the disk A
Substitute
Calculate the angular velocity of the disk B
Substitute
From Equation (5) and (6), it is clear that the final velocities of disk A and B are independent of coefficient of kinetic friction
(b)
Express the final angular velocity of disk A in terms of
Answer to Problem 16.44P
The final angular velocity of disk A in terms of
Explanation of Solution
The mass of the disk A is
The mass of the disk B is
The initial angular velocity of the disk A is
The coefficient of the kinetic friction is
The radius of the disk A is
The radius of the disk B is
The acceleration due to gravity is g.
The time required for the disk to come to rest is t.
Calculation:
Refer to part (a).
Calculate the final angular velocity of disk A in terms of
Refer Equation (5).
Hence, the final angular velocity of disk A in terms of
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