How Does Rotational Motion Affect a Cube and Pulley System?

[Superfluous]

A cube of mass M = 500 g and side length 30 mm is free to spin on an axis through the center of one face. A massless pulley on this axis has a diameter of 2r = 10 mm. A weight of m = 50 g is hung from a string wrapped around the pulley. The assembly is released from rest.

(a) Find the time to unwind L = 30 cm of string.

(b) Find the kinetic energy of the spinning block after the string unwinds, using energy methods.

I found a formula for time (I think):

t=\sqrt{\frac{2L}{r\alpha}}

I have L and r, but how do I find alpha? And also, how do I go about solving part b? I'm a bit confused on what I need to take into consideration.

 

[Doc Al]

I found a formula for time (I think):

t=\sqrt{\frac{2L}{r\alpha}}

Do you know how this formula was derived?

I have L and r, but how do I find alpha? And also, how do I go about solving part b? I'm a bit confused on what I need to take into consideration.

You find alpha by applying Newton's 2nd law to both bodies (the cube and the hanging weight) and solving for the acceleration. (You'll have to look up the rotational inertia for a cube.) Start by identifying the forces acting on each. How does linear acceleration of the falling mass relate to the angular acceleration of the pulley (and cube)?

For part b, use conservation of mechanical energy.