Concept explainers
Disk A has a mass mA = 4 kg, a radius rA = 300 mm, and an initial angular velocity ω0 = 300 rpm clockwise. Disk B has a mass mB = 1.6 kg, a radius rB = 180 mm, and is at rest when it is brought into contact with disk A. Knowing that μk = 0.35 between the disks and neglecting bearing friction, determine (a) the angular acceleration of each disk, (b) the reaction at the support C.
Fig. P16.43 and P16.44
(a)
Find the angular acceleration of each disk
Answer to Problem 16.43P
The angular acceleration of each disk
Explanation of Solution
The mass of the disk A
The mass of the disk B
The initial angular velocity of the disk A
The coefficient of the kinetic friction
The radius of the disk A
The radius of the disk B
Calculation:
Consider the acceleration due to gravity (g) as
Convert the unit of the radius of the disk A
Convert the unit of the radius of the disk B
Calculate the mass moment of inertia of the disk A
Substitute
Calculate the mass moment of inertia of the disk B
Substitute
Calculate the load of the disk A
Substitute
Calculate the load of the disk B
Substitute
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
Hence, the angular acceleration of each disk
(b)
Find the reaction at the support C
Answer to Problem 16.43P
The reaction at the support C
Explanation of Solution
The mass of the disk A
The mass of the disk B
The initial angular velocity of the disk A
The coefficient of the kinetic friction
The radius of the disk A
The radius of the disk B
Calculation:
Refer to part (a).
Show the free body diagram of the support C as in Figure 3.
Here,
Refer to Figure 3.
Calculate the horizontal forces by applying the equation of equilibrium:
Sum of horizontal forces is equal to 0.
Calculate the vertical forces by applying the equation of equilibrium:
Sum of vertical forces is equal to 0.
Calculate the moment about point C by applying the equation of equilibrium:
Sum of moments about point C is equal to 0.
Substitute
Calculate the time required for the disk to come to rest (t):
Substitute
Calculate the final angular velocity of the disk A
Substitute
Calculate the final angular velocity of the disk B
Substitute
Hence, the reaction at the support C
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