wo thin rectangular sheets (0.13 m 0.52 m) are identical. In the first sheet the axis of rotation lies along the 0.13 m side, and in the second it lies along the 0.52 m side. The same torque is applied to each sheet. The first sheet, starting from rest, reaches its final angular velocity in 8.6 s. How long does it take for the second sheet, starting from rest, to reach the same angular velocity?

The equation you need to use is T2=(T1*L1^2)/L2^2 Where T is the time , L is Axis of first sheet, l2 is axis of 2nd sheet

To determine the time it takes for the second sheet to reach the same angular velocity, we can use the principle of conservation of angular momentum.

The formula for angular momentum is L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.

Since the two sheets are identical except for the axis of rotation, they will have different moments of inertia (I). The moment of inertia depends on the mass and the distribution of mass around the axis of rotation.

However, since the torque applied to each sheet is the same, we know that the initial and final torques are the same for both sheets.

Using the formula τ = Iα, where τ is the torque and α is the angular acceleration, and assuming constant torque, we can find that τ = Iω₁/t₁ = Iω₂/t₂, where ω₁ and ω₂ are the initial and final angular velocities, and t₁ and t₂ are the times taken by the first and second sheets, respectively, to reach their final angular velocities.

Since the torque and moments of inertia are the same for both sheets, we can rearrange the equation to find t₂:

Iω₂ / t₂ = Iω₁ / t₁

Canceling out the moments of inertia and rearranging the equation gives:

ω₂ / t₂ = ω₁ / t₁

Now we can solve for t₂ by cross-multiplying:

t₂ = (ω₁ * t₁) / ω₂

Substituting the given values, where ω₁ is the final angular velocity of the first sheet (reached in 8.6 seconds), and ω₂ is the same angular velocity, we get:

t₂ = (ω₁ * 8.6) / ω₂

Thus, to find the time it takes for the second sheet to reach the same angular velocity, divide the product of the initial angular velocity (ω₁) and the time taken by the first sheet (t₁) by the final angular velocity (ω₂).