Concept explainers
(a)
Find the angular acceleration of each cylinder
(a)
Answer to Problem 16.45P
The angular acceleration of each cylinder
Explanation of Solution
The weight of the cylinder A
The weight of the cylinder B
The weight of the cylinder C
The initial angular velocity of the cylinder A
The coefficient of the kinetic friction
The radius of the cylinder A
The radius of the cylinder B
The radius of the cylinder C
Calculation:
Consider the acceleration due to gravity (g) as
Convert the unit of the radius of the cylinder A
Convert the unit of the radius of the cylinder B
Convert the unit of the radius of the cylinder C
Calculate the mass of the cylinder A
Substitute
Calculate the mass of the cylinder B
Substitute
Calculate the mass of the cylinder C
Substitute
Calculate the mass moment of inertia of the cylinder A
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Calculate the mass moment of inertia of the cylinder B
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Calculate the mass moment of inertia of the cylinder C
Substitute
Calculate the tangential acceleration of contact point between cylinder B and C
Here,
The friction force at the contact point between cylinder A and C
Show the free body diagram of the cylinder B as in Figure 1.
Here,
Refer to Figure 1.
Calculate the moment about point B by applying the equation of equilibrium:
Substitute
Show the free body diagram of the cylinder C as in Figure 2.
Here,
Refer to Figure 2.
Calculate the moment about point C by applying the equation of equilibrium:
Substitute
Calculate the normal force at the contact point between cylinder A and C
Substitute
Consider that the contact point between cylinder A and C is P.
Calculate the components of forces acting along the line CP:
Calculate the angular acceleration of the cylinder C
Substitute
Calculate the friction force at the contact point between cylinder A and C
Substitute
Calculate the friction force at the contact point between cylinder B and C
Substitute
Calculate the normal force at the contact point between cylinder A and C
Substitute
Show the free body diagram of the cylinder A as in Figure 3.
Here,
Refer to Figure 3.
Consider that the contact point between cylinder B and C is Q.
Calculate the components of forces acting along the line CQ:
Calculate the normal force at the contact point between cylinder B and C
Substitute
Calculate the angular acceleration of the cylinder A
Calculate the moment about point A by applying the equation of equilibrium:
Substitute
Calculate the angular acceleration of the cylinder A
Substitute
Hence, the angular acceleration of each cylinder
(b)
Find the final angular velocity of each disk
(b)
Answer to Problem 16.45P
The final angular velocity of each disk
Explanation of Solution
The weight of the cylinder A
The weight of the cylinder B
The weight of the cylinder C
The initial angular velocity of the cylinder A
The coefficient of the kinetic friction
The radius of the cylinder A
The radius of the cylinder B
The radius of the cylinder C
Calculation:
Refer to part (a).
The convert the unit of the initial angular velocity of the disk A
Calculate the angular velocity of cylinder A
Substitute
Calculate the tangential velocity of cylinder A
Substitute
Calculate the angular velocity of cylinder C
Substitute
Calculate the tangential velocity of cylinder C
Substitute
Calculate the time taken when tangential velocities are equal:
Substitute
Calculate the final angular velocity of the disk A
Substitute
Calculate the final angular velocity of the disk C
Substitute
Calculate the final angular velocity of the disk B
Substitute
Hence, the final angular velocity of each disk
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Chapter 16 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
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