Mechanics Physics: Pulley w/ 2 Blocks - Determine T, T3, Tl, I1

[princess_7777]

A pulley of radius R1 and rotational inertia I1 is mounted on an axle with negligible friction. A light cord passing over the pulley has two blocks of mass m attached to either end, as shown above. Assume that the cord does not slip on the pulley. Determine the answers to parts (a) and (b) in terms of m, R1, I1, and fundamental constants.
a. Determine the tension T in the cord.
b. One block is now removed from the right and hung on the left. When the system is released from rest, the three blocks on the left accelerate downward with an acceleration g/3 . Determine the following.
i. The tension T3 in the section of cord supporting the three blocks on the left
ii. The tension Tl in the section of cord supporting the single block on the right
iii. The rotational inertia I1 of the pulley

 

[hage567]

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[m2003]

a. To determine the tension T in the cord, we can use the equation for Newton's second law, F = ma. In this case, the force acting on the blocks is the tension in the cord, and the acceleration is the acceleration due to gravity, g. Therefore, T = mg, where m is the mass of each block.

b. When one block is removed from the right and hung on the left, the system becomes unbalanced and the blocks on the left will accelerate downwards. To determine the tensions in the cord, we can use the equations for Newton's second law again. The total force acting on the blocks on the left is the sum of the tensions in the cord, T3 and Tl, and the mass of the blocks, 3m (since there are now three blocks). Therefore, T3 + Tl = 3mg/3 = mg. We also know that the tension in the cord supporting the single block on the right is equal to the weight of that block, which is mg. So, Tl = mg.

To determine the rotational inertia of the pulley, we can use the equation for the rotational equivalent of Newton's second law, τ = Iα. In this case, the torque acting on the pulley is the tension in the cord, T, multiplied by the radius of the pulley, R1. The acceleration is the angular acceleration, α, which is related to the linear acceleration of the blocks, a, by α = a/R1. Therefore, T*R1 = I*(a/R1). Rearranging this equation, we get I = T*R1*a. We know the values of T and a from part (a), so we can calculate I in terms of m, R1, and fundamental constants.