How Does Frictionless Pulley Affect Acceleration in a Modified Atwood's Machine?

[Vixenbergen]

Homework Statement

Given the system shown, if the pulley has frictionless bearings, find
a, the magnitude of the acceleration of the masses.

This is a modified Atwood's machine, instead of the 2nd mass hanging however, it is stationary on a table. The pulley is at the edge of the table and the 1st mass is hanging down. The mass on the table has a mass of 3kg, the hanging mass is 4kg. The pulley has an I of 1/2 kg*m^2 and a radius of .3m.

Homework Equations

F = Ma
T = Ia(angular)
T = r x F

The Attempt at a Solution

F = Ma
Mg - T2 = ma
also Torque = r x F
Torque = rTsin90
RT = Ia(angular)
rT = I (A/R) --> because linear accel = angular * radius

But from here I'm lost...

 

[phinds]

Given the system shown

? There is no system shown. Did you forget your diagram?

 

[Delta2]

The critical point here is that the tension of the rope is not equal between the two sides of the pulley because the pulley has a moment of inertia hence it needs torque to rotate, and the torque would be zero if the two tensions were equal. So we have to use ##T_1,T_2## for the two tensions, and the system of equations is (by index 1 I mean the mass that is hanging)
$$m_1g-T_1=m_1a$$
$$T_2=m_2a$$
$$(T_1-T_2)R=I\frac{a}{R}$$

Linear system of three equations with three unknowns, shouldn't be much of a trouble to solve for a college student.

 

[kuruman]

? There is no system shown. Did you forget your diagram?

I think that here we have what is more widely known as a "half-Atwood" machine.