Two thin rectangular sheets (0.29 m 0.40 m) are identical. In the first sheet the axis of rotation lies along the 0.29-m side, and in the second it lies along the 0.40-m side. The same torque is applied to each sheet. The first sheet, starting from rest, reaches its final angular velocity in 7.7 s. How long does it take for the second sheet, starting from rest, to reach the same angular velocity?
Moment of intertia is proportional to the square of the dimension perpendicular to the axis of rotation. It is therefore (40/29)^2 = 1.902 times higher when the axis is the 0.29 m side. That would be the first sheet.
When the same torque is applied, the angular acceleration in inversely proportional to moment of inertia.
Angular velocity is proportional to the angular acceleration.
The second sheet wil take 1/1.902 times as long to attain the same angular velocity as the first.
To find the time it takes for the second sheet to reach the same angular velocity as the first sheet, we can use the principle of conservation of angular momentum.
The angular momentum of an object is given by the product of its moment of inertia (I) and its angular velocity (ω). Since the sheets are identical, their moment of inertia will be the same.
We can write the equation for angular momentum as:
L = I * ω
For the first sheet, the axis of rotation lies along the 0.29 m side. The length of the side along the axis of rotation corresponds to the moment of inertia in this case. Let's call it I₁.
For the second sheet, the axis of rotation lies along the 0.40 m side. The length of the side along the axis of rotation corresponds to the moment of inertia in this case. Let's call it I₂.
Since the sheets are identical, I₁ = I₂ = I (let's call this common moment of inertia).
Now, the torque equation can also be written as:
τ = I * α
where τ is the torque applied to the sheet and α is the angular acceleration.
Since the same torque is applied to both sheets, we can equate their torque equations:
I₁ * α = I₂ * α
Since I₁ = I₂ = I, we have:
I * α = I * α
This means that both sheets experience the same angular acceleration.
Given that the first sheet reaches its final angular velocity in 7.7 s, we can write the equation for angular velocity:
ω = α * t
where ω is the angular velocity, α is the angular acceleration, and t is the time.
For the first sheet:
ω₁ = α * t₁ (where ω₁ is the angular velocity of the first sheet and t₁ is the time taken for the first sheet to reach its final angular velocity)
For the second sheet:
ω₂ = α * t₂ (where ω₂ is the angular velocity of the second sheet and t₂ is the time taken for the second sheet to reach its final angular velocity)
Since both sheets experience the same angular acceleration, we can write:
ω₁ = ω₂
α * t₁ = α * t₂
Dividing both sides by α:
t₁ = t₂
This means that the time taken for the second sheet to reach the same angular velocity as the first sheet is also 7.7 s.