Two thin rectangular sheets (0.09 m 0.46 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 0.46-m side. The same torque is applied to each sheet. The first sheet, starting from rest, reaches its final angular velocity in 7.5 s. How long does it take for the second sheet, starting from rest, to reach the same angular velocity?

Well, isn't that a twist! The first sheet reaches its final angular velocity in 7.5 seconds. Now, for the second sheet, we switch things up and the axis of rotation lies along the longer side.

Now, before we get too twisted up in knots, remember that the moment of inertia depends on the axis of rotation. The moment of inertia for a rectangular sheet rotating about the shorter side is different from rotating about the longer side.

But fear not, my friend! We can still solve this puzzle. Since the sheets are identical, we can use the principle of conservation of angular momentum. This means that the angular momenta of the two sheets are equal.

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 torque applied to each sheet is the same, the initial angular momentum is the same for both sheets.

Therefore, we can write:

L1 = L2

I1ω1 = I2ω2

0.09*m1*ω1 = 0.46*m2*ω2

Now we are comparing the moment of inertia for the first sheet (I1) to the moment of inertia for the second sheet (I2). The masses (m1, m2) cancel out because the sheets are identical.

So, we have:

0.09*ω1 = 0.46*ω2

Now we can solve for ω2, which is the angular velocity of the second sheet:

ω2 = (0.09/0.46)*ω1

ω2 = 0.1957*ω1

So, the angular velocity of the second sheet is 0.1957 times the angular velocity of the first sheet.

Now, to find out how long it takes for the second sheet to reach the same angular velocity as the first sheet, we just apply the good ol' formula of time = (change in angular velocity) / (angular acceleration).

But since we started from rest, the initial angular velocity is 0, so the change in angular velocity is just ω2. And since we applied the same torque to both sheets, the angular acceleration is the same for both sheets.

Therefore, the time it takes for the second sheet to reach the same angular velocity as the first sheet is:

time = ω2 / angular acceleration

But since the angular acceleration is the same for both sheets, the time it takes for the second sheet is the same as the time it took for the first sheet, which is 7.5 seconds.

So, in the grand finale, the second sheet takes 7.5 seconds to reach the same angular velocity as the first sheet. Ta-da!

To find the time it takes for the second sheet to reach the same angular velocity, we can use the concept of rotational inertia.

The rotational inertia, also known as the moment of inertia, depends on both the mass and the distribution of mass around an axis of rotation. For a thin rectangular sheet rotating about an axis perpendicular to its sides, the rotational inertia is given by the formula:

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

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

In this case, since the sheets are identical, the rotational inertia for both sheets will be the same.

Let's assume the mass of each sheet is m, and the dimensions are given as 0.09 m and 0.46 m.

For the first sheet, the axis of rotation lies along the 0.09 m side. So, the rotational inertia for the first sheet (I1) can be calculated as:

I1 = (1/12) * m * (0.09^2 + 0.46^2)

Similarly, for the second sheet, the axis of rotation lies along the 0.46 m side. So, the rotational inertia for the second sheet (I2) can be calculated as:

I2 = (1/12) * m * (0.46^2 + 0.09^2)

Since the applied torque is the same for both sheets, the angular acceleration will be the same. We can use the following formula to relate torque (τ), rotational inertia (I), and angular acceleration (α):

τ = I * α

Since the torque is the same, and the rotational inertia is the same for both sheets, the angular acceleration will also be the same.

We know that the first sheet reaches its final angular velocity in 7.5 s. Let's assume this angular velocity is ω1. We can use the following formula to relate angular velocity (ω), angular acceleration (α), and time (t):

ω = α * t

For the first sheet,

ω1 = α * 7.5

For the second sheet, let's assume the time it takes to reach the same angular velocity is t2. We can use the same formula to relate angular velocity and time:

ω2 = α * t2

Since the angular acceleration is the same for both sheets, we can equate the angular velocities:

ω1 = ω2

α * 7.5 = α * t2

Simplifying the equation, the angular acceleration cancels out:

7.5 = t2

Therefore, it takes 7.5 seconds for the second sheet, starting from rest, to reach the same angular velocity as the first sheet.

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 (L) is given by the equation L = Iω, where I is the moment of inertia and ω is the angular velocity.

Since the two sheets are identical except for the axis of rotation, the moment of inertia for each sheet can be calculated using the formula for the moment of inertia of a rectangle rotating about one of its sides:

I = (1/12) * (mass * (length^2 + width^2))

Given that the two sheets are identical, their masses and dimensions are the same. So we can assume the mass and dimensions of one sheet for simplicity.

Let's calculate the moment of inertia for the first sheet:

First sheet dimensions: length = 0.09 m, width = 0.46 m

I1 = (1/12) * (m * (0.09^2 + 0.46^2))

Now, we know that the torque applied to each sheet is the same.

The torque (τ) is given by the equation τ = Iα, where α is the angular acceleration.

Since the torque is the same for both sheets and assuming the same mass, we have:

τ1 = I1 * α1

τ2 = I2 * α2

Since τ1 = τ2 and I1 = I2, we have:

α1 = α2

Now, let's find the relationship between the angular acceleration (α) and time (t) using the equation:

ω = α * t

For the first sheet, ω1 = α1 * t1, where t1 = 7.5 s.

For the second sheet, we want to find t2.

ω2 = α2 * t2

Since α1 = α2, we can write:

ω1 = α1 * t1 = α2 * t1 = ω2

So, t2 = ω1 / α2 = t1 = 7.5 s

Therefore, it takes the second sheet the same 7.5 seconds to reach the same angular velocity as the first sheet.

The moment of inertia of the rectangular plate about its side is

I =ma²/3 , where a is the length of another side (perpendicular to the axis of rotation), therefore,
I1 =m•(0.46)²/3,
I2 =m•(0.09)²/3,

Final angular velocity is
ω = ε1•t1, ε1 =ω /t1
ω = ε2•t2, • ε2 =ω /t2

The Newton’s 2 Law for rotation
M=I•ε,
M=I1• ε1 = I1• ω /t1= m•(0.46)² • ω /3•t1,
M=I2• ε2= I2• ω /t2 = m•(0.09)² • ω /3•t2,
m•(0.46)² • ω /3•t1 = m•(0.09)² • ω /3•t2,
(0.46)² /t1 =(0.09)² /t2,
t2 =t1• (0.09/0.46)² = 7.5•3.8•10^-2= 0.28 s.