Calculate the angular acceleration of the pulleys, Mechanical Engineering

Calculate the angular acceleration of the pulleys:

The compound pulley system has a mass of 30 kg and a radius of gyration of 450 mm. Calculate the tension in each cord and the angular acceleration of the pulleys when the masses are released (Refer following Figure).

1648_angular acceleration of the pulleys.jpg


Free body diagram for the pulley and both the masses are drawn separately as shown in the figure.

We must keep in mind that a1 = 0.6 α and a2 = 0.3 α acceleration (i.e. difference in radii). where α is the angular

 Inertial forces are illustrated on FBD. Now, we may write down dynamic equilibrium equations for each of the body.

For a mass of 50 kg, we have

∑ Fy  = 0                     ∴ T1 - 50 a1 - 50 × 9.81 = 0

Substituting for a1 in terms of α, we get

T1  = 50 × 0.6 α + 50 × 9.81

= 490.5 + 30 α  ------------- (1)

For the mass of 150 kg, we have

∑ Fy  = 0                  ∴ T2  + 150 . a2  - 150 × 9.81 = 0

∴ T2  = 1471.5 - 150 × 0.3 α

= 1471.5 - 45 α             ---------- (2)   

For the pulley, we have ∑ M  = 0

∴ 0.6 T1  + I α - 0.3 T2  = 0

But       I = 30 × (0.45)2  = 6.08 kg m 2

∴ 0.6 T1  - 0.3 T2  = - 6.08 α                  ---------- (3)

Solving Eqs. (1), (2) and (3) as below, we obtain

0.6 T1  - 0.3 T2  = - 6.08 α

0.6 (490.5 + 30 α) - 0.3 (1471.5 - 45 α) = - 6.08 α

or,        147.15 = 37.58 α

α = 3.91 rad / sec.

Substituting this value of α in Eq. (1) and (2) above we obtain

T1  = 607.8 N

and T2  = 1295.55 N.

Posted Date: 1/29/2013 6:36:25 AM | Location : United States

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