Question

Two blocks hang from either end of a massless rope that runs over a pulley, treated as a thin solid disk, (An Atwood's Machine), and are held in place. One block has a mass of 12.0 kg, the pulley has a mass of 2.00 kg and radius 5.00 cm, and the other block's mass is unknown. The blocks are released from rest, and after an unspecified period of time, the block of known mass has descended 2.50 m and has a velocity of 3.00 m/s toward the ground. The rope does not slip over the pulley, and the pulley spins on a frictionless axle. Find the unknown mass of the block

Answer 1

initial velocity , u = 0

after displacement, r = 2.50 m

final velocity, v = 3 m/s

Applying v^2 - u^2 = 2 a r

3^2 - 0^2 = 2 x a x 2.50

a = 1.8 m/s^2

so know mass is heavier.

ON block, Applying Fnet = m a

12g - T1 = 12 a .............(i)

on pulley, { net torque = I x alpha and alpha = a / R}

R(T1 - T2) = (M R^2 / 2) (a/R)

T1 - T2 = M a / 2 = a ...........(ii)

Now for unkown mass:

T2 - mg = m a ...........(iii)

i + ii + iii =>

12g - mg = (12 + 1 + m) a

and a = 1.8 m/s^2

12 x 9.8   - 9.8m = 13 x 1.8 + 1.8m

m = 8.12 kg .........Ans