Two thin rectangular sheets (0.09 m 0.54 m) are identical. In the first sheet the axis of rotation lies along the 0.09-m side, and in the second it lies along the 0.54-m side. The same torque is applied to each sheet. The first sheet, starting from rest, reaches its final angular velocity in 8.4 s. How long does it take for the second sheet, starting from rest, to reach the same angular velocity?

What is your school subject?

This looks like physics, not first aid.

The rate of angular acceleration is proportional to torque and inversely proportional to the moment of inertia.

It is not clear from your question how the axis of rotation is aligned with the sheet.

To solve this problem, let's first understand the concept of torque and angular velocity.

Torque (τ) is the rotational equivalent of force and is given by the formula τ = r x F, where r is the distance from the axis of rotation and F is the applied force. Torque can also be written as τ = Iα, where I is the moment of inertia and α is the angular acceleration.

Angular velocity (ω) is the rate at which an object rotates and is given by the formula ω = Δθ / Δt, where Δθ is the change in angle and Δt is the time taken.

In this problem, both sheets are identical in size but have different orientations. The first sheet has an axis of rotation along the 0.09 m side, while the second sheet has an axis of rotation along the 0.54 m side.

We are told that the same torque is applied to each sheet and that the first sheet reaches its final angular velocity in 8.4 seconds. We need to find how long it takes for the second sheet to reach the same angular velocity.

Let's use the concept of moment of inertia (I) to solve this problem. Moment of inertia depends on the shape and distribution of mass in an object. For a thin rectangular sheet rotating about an axis perpendicular to its length, the moment of inertia is given by I = 1/12 * m * L^2, where m is the mass and L is the length of the side along the axis of rotation.

Since the sheets are identical, their moment of inertia (I) will be the same. However, the lengths of the sides along the axis of rotation are different. For the first sheet, L = 0.09 m, and for the second sheet, L = 0.54 m.

We are given the time it takes for the first sheet to reach its final angular velocity, so we can use the formula ω = α * t to find the angular acceleration (α). Rearranging the formula gives us α = ω / t.

Using the moment of inertia (I), torque (τ), and angular acceleration (α), we can write the equation τ = I * α.

Now, let's find the angular acceleration for the first sheet. We know the final angular velocity (ω) is reached in 8.4 seconds, so ω = Δθ / Δt. Assuming the starting angle is zero, we have ω = Δθ / 8.4. Rearranging the formula gives us Δθ = ω * 8.4.

We can substitute the value of Δθ in terms of ω into the equation for α to get α = ω / Δt = ω / 8.4.

With α known, we can use the equation τ = I * α to find the torque applied to each sheet.

Now, let's find the time it takes for the second sheet to reach the same angular velocity. We have the torque (τ) applied to the second sheet and the moment of inertia (I) is the same for both sheets. Using the formula τ = I * α, we can solve for α.

Once we have α, we can use the formula α = Δθ / Δt to find the time it takes for the second sheet to reach the same angular velocity. Rearranging the formula gives us Δt = Δθ / α.

Substitute the known values into the equation, and you will find the time it takes for the second sheet to reach the same angular velocity.