How to Determine the Velocity of Block A with Pulley Masses Included?

[That_EDGEy_KiD]

Homework Statement

Block A has a mass of 3 kg, and block B has a mass of 8 kg. Determine the speed of block A if it moves upwards 2 meters, starting from rest. I can solve the problem pretty easily if the mass and radius of each of the pulleys is neglected. However, if they are not neglected and let's say that the question provides a known mass and radius for each of the top and bottom pulleys, how would I still go about solving for the velocity of block A?

Homework Equations

KE = (1/2)mv^2
g = 9.8 m/s^2

The Attempt at a Solution

Neglecting the mass/radius of the pulleys,

(1/2)mAvA^2 + (1/2)mB(2vB)^2 = mBg(4) - mAg(2)

=> 17.5vA^2 = (8x4 - 3x2)g
=> 17.5vA^2 = 26g
=> vA = 3.82 m/s

However, If the top pulley had a mass of say 0.50 kg with a radius of 0.10 m and the bottom pulley had a mass of say 0.40 kg with a radius of 0.5 m, how would I solve for the velocity of block A since now the tensions are all different?

 

[drvrm]

However, If the top pulley had a mass of say 0.50 kg with a radius of 0.10 m and the bottom pulley had a mass of say 0.40 kg with a radius of 0.5 m, how would I solve for the velocity of block A since now the tensions are all different?

start the attempt by drawing a free body diagram and write down the net force working on masses/pulleys- as bodies are connected displacements are related and with their proper sign and apply the time rate of change of displacements to get velocity and in turn accelerations.
pully with masses may not hold the energy conservation unless stated

 

[haruspex]

if they are not neglected and let's say that the question provides a known mass and radius for each of the top and bottom pulleys, how would I still go about solving for the velocity of block A?

In that situation, you can no longer assume tension is constant along a rope. Allow for a different tension on each stretch and consider the net torque and resulting angular acceleration of each pulley. Relate these angular accelerations to the linear accelerations of the masses using the rolling contact condition.

 

[haruspex]

pully with masses may not hold the energy conservation unless stated

There's nothing special about pulleys with mass in that regard. It's just an extra place for kinetic energy to be accumulated. Typically in pulley questions, massless or otherwise, you can assume work conservation, and indeed need to to solve it. An exception is when frictional torque is mentioned, or when there is enough information to deduce the magnitude of frictional torque.