How Long Will It Take the Second Sheet to Reach the Same Angular Velocity?

[PhysicsZax]

Homework Statement

Two thin rectangular sheets (0.20 m x 0.43 m) are identical. In the first sheet the axis of rotation lies along the 0.20-m side, and in the second it lies along the 0.43-m side. The same torque is applied to each sheet. The first sheet, starting from rest, reaches its final angular velocity in 8.8 s. How long does it take for the second sheet, starting from rest, to reach the same angular velocity?

Homework Equations

T = Iw/t

The Attempt at a Solution

The moment of inertia for a sheet of paper, I = 1/3mL^2

((0.43)^2 * 1/3) / 8.8 s = ((0.2)^2 * 1/3) / t

t = 1.9 s

This doesn't appear to be the correct answer. Any ideas? Do I have the Lengths backwards? Any help would be much appreciated!

Thanks!

Zach

 

[anathema]

Dear Zach,

Thank you for your post. Your attempt at a solution is on the right track, but there are a few things to consider.

Firstly, in the equation T = Iw/t, T represents torque, I represents moment of inertia, w represents angular velocity, and t represents time. So, when you rearrange the equation to solve for t, it should be t = Iw/T.

Secondly, the moment of inertia for a rectangular sheet is given by I = (1/12)ml^2, where m is the mass and l is the length of the sheet. In this case, the sheets are identical, so their masses are the same. Therefore, the moment of inertia for both sheets will be the same.

Now, let's apply these concepts to the problem. We know that the moment of inertia, I, is the same for both sheets. We also know that the torque, T, is the same for both sheets. The only difference between the two sheets is the length, l, along which the axis of rotation lies. So, we can set up the following equation:

((0.2)^2 * 1/12) * w / T = ((0.43)^2 * 1/12) * w / t

Simplifying, we get:

t = (0.43)^2 / (0.2)^2 * 8.8 s

t = 2.35 s

So, it will take the second sheet 2.35 seconds to reach the same angular velocity as the first sheet.

I hope this helps! Keep up the good work.

 

[ogb p]

Your calculation is correct, but it seems that you have mixed up the lengths for the two sheets. In the first sheet, the axis of rotation lies along the 0.20-m side, so the length used in the moment of inertia calculation should be 0.20 m. In the second sheet, the axis of rotation lies along the 0.43-m side, so the length used should be 0.43 m. Therefore, the correct equation would be:

((0.20)^2 * 1/3) / 8.8 s = ((0.43)^2 * 1/3) / t

Solving for t, we get t = 4.8 s. This is the time it would take for the second sheet, starting from rest, to reach the same angular velocity as the first sheet.