When this block is released and so is free to move, what is its speed after it h

Question

When this block is released and so is free to move, what is its speed after it has risen a height h? Express your answer in terms of some or all of the variables m, R, h, and the acceleration due to gravity g.

Problem 11.66 A block of inertia m is attached to a block of inertia 2m by a light string draped over a uniform disk of inertia 3m and radius R that can rotate on a horizontal axle (Figure 1). Ignore friction between disk and axle. Initially, the block of inertia m is held in place so that the string is taut. Figure 1 of 1 3m 2m

Explanation / Answer

let a be the accleration of the masses

and A be the angular acceration of the puley

aldo T2 and T1 are the tensions in the strings

Now

2mg -T2 = 2 ma ---------------------------1

T1 - mg ma ----------------------------------2
also a and A are related by a = R A ---------------------3

(T2-T1) *R I A = 0.5(3M) R^2 A

(T2-T1) R = 3MR^2a/2R

T2-T1 = 3ma/2

so

2m2g - T2 + T1 -mg = 3ma

T2-T1 = mg - 3ma

9 a/2 = g

a = 2g/9

finally use V^2 - u^2 = 2a s

v = 2/3 sqrt(gh)

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