Concept explainers
Each of the double pulleys shown has a mass moment of inertia of 15 lb·ft·s2 and is initially at rest. The outside radius is 18 in., and the inner radius is 9 in. Determine (a) the angular acceleration of each pulley, (b) the angular velocity of each pulley after point A on the cord has moved 10 ft.
Fig. P16.34
(a)
Find the angular acceleration of the pulley 1
Find the angular acceleration of the pulley 2
Find the angular acceleration of the pulley 3
Find the angular acceleration of the pulley 4
Answer to Problem 16.34P
The angular acceleration of the pulley 1
The angular acceleration of the pulley 2
The angular acceleration of the pulley 3
The angular acceleration of the pulley 4
Explanation of Solution
The mass moment of inertia of the double pulleys
The outside radius of the pulley
The inner radius of the pulley
The finial angular velocity of the pulley
The load of the pulley 1
The load of the pulley 2
The left side load of the pulley 3
The right side load of the pulley 3
The load of the pulley 4
Calculation:
Consider the acceleration due to gravity (g) is
Case 1:
Convert the unit of the outside radius
Convert the unit of the inner radius
Show the free body diagram of the double pulley 1 as in Figure 1.
Here,
Refer to Figure 1.
Calculate the angular acceleration of the pulley 1
Calculate the moment about point O by applying the equation of equilibrium:
Hence, the angular acceleration of the pulley 1
Case 2:
Calculate the mass of the pulley 2
Substitute
Show the free body diagram of the double pulley 2 as in Figure 2.
Refer to Figure 2.
Calculate the moment about point O by applying the equation of equilibrium:
Calculate the angular acceleration of the pulley 2
Substitute
Hence, the angular acceleration of the pulley 2
Case 3:
Calculate the left side mass of the pulley 3
Substitute
Calculate the right side mass of the pulley 3
Substitute
Show the free body diagram of the double pulley 3 as in Figure 3.
Refer to Figure 3.
Calculate the moment about point O by applying the equation of equilibrium:
Calculate the angular acceleration of the pulley 3
Substitute
Hence, the angular acceleration of the pulley 3
Case 4:
Calculate the left side mass of the pulley 4
Substitute
Show the free body diagram of the double pulley 4 as in Figure 4.
Refer to Figure 4.
Calculate the moment about point O by applying the equation of equilibrium:
Calculate the angular acceleration of the pulley 4
Substitute
Hence, the angular acceleration of the pulley 4
(b)
Find the angular velocity of the pulley 1
Find the angular velocity of the pulley 2
Find the angular velocity of the pulley 3
Find the angular velocity of the pulley 4
Answer to Problem 16.34P
The angular velocity of the pulley 1
The angular velocity of the pulley 2
The angular velocity of the pulley 3
The angular velocity of the pulley 4
Explanation of Solution
The mass moment of inertia of the double pulleys
The outside radius of the pulley
The inner radius of the pulley
The finial angular velocity of the pulley
The load of the pulley 1
The load of the pulley 2
The left side load of the pulley 3
The right side load of the pulley 3
The load of the pulley 4
The moved distance of the point A (l) is
Calculation:
Refer part (a).
Case 1:
Calculate the angle of the pulley 1
Substitute
Calculate the angular velocity of the pulley 1
Substitute
Hence, the angular velocity of the pulley 1
Case 2:
Calculate the angle of the pulley 2
Substitute
Calculate the angular velocity of the pulley 2
Substitute
Hence, the angular velocity of the pulley 2
Case 3:
Calculate the angle of the pulley 3
Substitute
Calculate the angular velocity of the pulley 3
Substitute
Hence, the angular velocity of the pulley 3
Case 4:
Calculate the angle of the pulley 4
Substitute
Calculate the angular velocity of the pulley 4
Substitute
Hence, the angular velocity of the pulley 4
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