Two thin rectangular sheets (0.22 m 0.35 m) are identical. In the first sheet the axis of rotation lies along the 0.22-m side, and in the second it lies along the 0.35-m side. The same torque is applied to each sheet. The first sheet, starting from rest, reaches its final angular velocity in 9.5 s. How long does it take for the second sheet, starting from rest, to reach the same angular velocity?

To solve this problem, we can use the concept of rotational kinematics. In rotational motion, the angular acceleration (α), final angular velocity (ω), and time (t) are related by the equation:

ω = α * t

Given that the first sheet reaches its final angular velocity (ω) in 9.5 seconds, we can determine the angular acceleration (α) by dividing the angular velocity (ω) by the time (t):

α = ω / t = ω / 9.5

Now, we need to find the time (t') it takes for the second sheet to reach the same angular velocity as the first sheet. Since the torque applied to each sheet is the same, the angular acceleration (α') of the second sheet will also be the same. Thus, we can use the equation:

ω' = α' * t'

Where ω' is the final angular velocity of the second sheet and α' is the angular acceleration.

To find ω', we can use the fact that the second sheet is identical to the first sheet, except that the axis of rotation is along the 0.35 m side instead of the 0.22 m side. The second sheet has a different moment of inertia due to this difference in the axis of rotation.

The moment of inertia (I) varies depending on the axis of rotation, but it can be related to the mass (m) and dimensions of an object using formulas specific to the shape and axis of rotation. In this case, the sheets are thin rectangular sheets, and we need to use the formula for the moment of inertia of a rectangular plate rotating about an axis perpendicular to the plate and passing through its center:

I = (1/12) * m * (a^2 + b^2)

Where m is the mass of the sheet, and a and b are the dimensions of the sheet.

Since the sheets are identical, the mass and material are the same. Therefore, the only difference in the moment of inertia (I') of the second sheet is due to the difference in dimensions.

Let's calculate the moment of inertia for both sheets:

For the first sheet:
I1 = (1/12) * m * (0.22^2 + 0.35^2)

For the second sheet:
I2 = (1/12) * m * (0.35^2 + 0.22^2)

Now, since the torque applied to each sheet is the same, we can use the torque equation to relate the moment of inertia (I'), the angular acceleration (α'), and the torque (τ):

τ = I' * α'

Since τ is the same for both sheets, I' and α' must be related as well:

I1 * α' = I2 * α'

Now we can solve for α':

α' = (I1 * α') / I2

We know that α' = α, so we can substitute α' = (ω / 9.5) into the equation:

(ω / 9.5) = (I1 * α') / I2

Now, we can solve for ω', the final angular velocity of the second sheet:

ω' = (ω * I1 * I2) / (9.5 * I2 + 9.5 * I1)

Finally, to find the time (t') it takes for the second sheet to reach the same angular velocity as the first sheet, we can rearrange the equation:

t' = ω' / α' = ω' / (ω / 9.5)

t' = (ω' * 9.5) / ω

By substituting the values of ω' and ω into the equation, we can calculate the time (t') it takes for the second sheet to reach the same angular velocity as the first sheet.