Concept explainers
A belt of negligible mass passes between cylinders A and B and is pulled to the right with a force P. Cylinders A and B weigh, respectively, 5 and 20 lb. The shaft of cylinder A is free to slide in a vertical slot and the coefficients of friction between the belt and each of the cylinders are μs = 0.50 and μk = 0.40. For P = 3.6 lb, determine (a) whether slipping occurs between the belt and either cylinder, (b) the angular acceleration of each cylinder.
Fig. P16.39
(a)
Find whether slipping occurs between the belt and either cylinder.
Explanation of Solution
The force pulled between cylinders A and B (P) is
The weight of the cylinder A
The weight of the cylinder B
The coefficient of the static friction
The coefficient of the kinetic friction
The radius of the cylinder A
The radius of the cylinder B
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
Consider that no slipping occurs.
Calculate the acceleration of the belt
Calculate the mass of the cylinder A
Substitute
Calculate the mass of the cylinder B
Substitute
Calculate the mass moment of inertia of the cylinder A
Substitute
Calculate the mass moment of inertia of the cylinder B
Substitute
Show the free body diagram of the cylinder A as in Figure 1.
Here,
Refer to Figure 1.
Calculate the moment about point G by applying the equation of equilibrium:
Substitute
Show the free body diagram of the cylinder B as in Figure 2.
Here,
Refer to Figure 2.
Calculate the moment about point G by applying the equation of equilibrium:
Substitute
Show the free body diagram of the belt as in Figure 3.
Refer to Figure 3.
Calculate the horizontal forces by applying the equation of equilibrium:
Sum of horizontal forces is equal to 0.
Calculate the angular acceleration of the cylinder A
Substitute
Calculate the horizontal force of the cylinder A
Substitute
Calculate the horizontal force of the cylinder B
Substitute
Calculate the magnitude of the friction force
Substitute
The horizontal force of the cylinder B is greater than the magnitude of the friction force
Therefore, the slipping occurs between cylinder B and the belt and the slipping not occur between cylinder A and the belt.
(b)
Find the angular acceleration of each cylinder
Answer to Problem 16.39P
The angular acceleration of each cylinder
Explanation of Solution
The force pulled between cylinders A and B (P) is
The weight of the cylinder A
The weight of the cylinder B
The coefficient of the static friction
The coefficient of the kinetic friction
The radius of the cylinder A
The radius of the cylinder B
Calculation:
Refer the part (a).
Consider the slipping occurs at cylinder B.
Therefore, the angular acceleration of the cylinder B is
Calculate the horizontal force of the cylinder B
Substitute
Show the free body diagram of the cylinder B as in Figure 4.
Here,
Refer to Figure 4.
Calculate the angular acceleration of the cylinder B
Calculate the moment about point G by applying the equation of equilibrium:
Substitute
Calculate the horizontal force of the cylinder A
Substitute
The horizontal force of the cylinder A is less than the force of the cylinder B due to the static friction
There is no slipping between the cylinder A and the belt.
Calculate the angular acceleration of the cylinder A
Substitute
Hence, the angular acceleration of each cylinder
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