Angular velocity and acceleration of a plank

[Kevodaboss]

Homework Statement

A horizontal plank with a frictionless axis of rotation at its center has a large mass at one end and a small mass at the other end. It is held stationary and then released from rest. As the plank rotates (and before one end hits the ground), the magnitude of the angular acceleration of the plank (increases/decreases/or remains constant) and the magnitude of the angular velocity (increases/decreases/or remains constant) ?

Homework Equations

τ=I α
τ=r F_perp

The Attempt at a Solution

The solution to this problem is that τ=I α, therefore if torque decreases, then angular acceleration increases. Thus the magnitude angular velocity must be increasing, because the angular acceleration is nonzero.

However, I can't seem to figure out why the torque is decreasing in this problem. I know that τ=r F_perp, but how can I apply this to the problem? The lever arm obviously stays the same length. So is the perpendicular force decreasing because after the plank tilts downwards, the force of gravity downwards has both a horizontal component and a vertical component with respect to the plank (force of gravity stays the same)?

Thanks for reading! Any help is appreciated

 

[Doc Al]

The lever arm obviously stays the same length.

Usually the "lever arm" is defined as the perpendicular distance from the line of force to the axis of rotation. So by that definition, the lever arm decreases.

The length of the plank doesn't change, so I suspect that's what you meant.

So is the perpendicular force decreasing because after the plank tilts downwards, the force of gravity downwards has both a horizontal component and a vertical component with respect to the plank (force of gravity stays the same)?

Right. The component of the weight perpendicular to the plank decreases, so the torque decreases.

 

[Kevodaboss]

Ok, that makes sense. Thanks for the help!

Usually the "lever arm" is defined as the perpendicular distance from the line of force to the axis of rotation. So by that definition, the lever arm decreases.

The length of the plank doesn't change, so I suspect that's what you meant.

Right. The component of the weight perpendicular to the plank decreases, so the torque decreases.

 

[CWatters]

The Attempt at a Solution

The solution to this problem is that τ=I α, therefore if torque decreases, then angular acceleration increases.

Check that.

 

[Kevodaboss]

Check that.

What do you mean?

 

[Kevodaboss]

Check that.

Oops never mind. I meant if torque decreases, angular acceleration decreases

 

[Doc Al]

Oops never mind. I meant if torque decreases, angular acceleration decreases

I had assumed you just messed up that sentence, since what followed didn't depend on it. (Otherwise I would have said something.)

 

[Kevodaboss]

I had assumed you just messed up that sentence, since what followed didn't depend on it. (Otherwise I would have said something.)

Yeah, just a typo. Thanks for all the help!