To solve the problem, we need to find the torque required to stop the rotation of a wheel with a given moment of inertia in a specified time. Here’s the step-by-step solution:
### Step 1: Understand the relationship between torque and angular momentum
Torque (τ) is related to the change in angular momentum (ΔL) over time (Δt) by the formula:
\[
\tau = \frac{\Delta L}{\Delta t}
\]
### Step 2: Identify initial and final angular momentum
The angular momentum (L) of a rotating object is given by:
\[
L = I \cdot \omega
\]
where:
- \(I\) is the moment of inertia,
- \(\omega\) is the angular velocity.
Given:
- Moment of inertia \(I = 4 \, \text{kg m}^2\),
- Initial angular velocity \(\omega_i = 240 \, \text{rpm}\).
### Step 3: Convert angular velocity from rpm to rad/s
To convert revolutions per minute (rpm) to radians per second (rad/s):
\[
\omega_i = 240 \, \text{rpm} \times \frac{2\pi \, \text{rad}}{1 \, \text{revolution}} \times \frac{1 \, \text{minute}}{60 \, \text{seconds}} = 240 \times \frac{2\pi}{60} = 8\pi \, \text{rad/s}
\]
### Step 4: Calculate initial and final angular momentum
- Initial angular momentum \(L_i = I \cdot \omega_i = 4 \, \text{kg m}^2 \cdot 8\pi \, \text{rad/s} = 32\pi \, \text{kg m}^2/\text{s}\).
- Final angular momentum \(L_f = 0\) (since the wheel stops).
### Step 5: Find the change in angular momentum
\[
\Delta L = L_f - L_i = 0 - 32\pi = -32\pi \, \text{kg m}^2/\text{s}
\]
### Step 6: Determine the time interval
The time interval given is 1 minute, which is:
\[
\Delta t = 60 \, \text{seconds}
\]
### Step 7: Calculate the torque
Now we can substitute the values into the torque formula:
\[
\tau = \frac{\Delta L}{\Delta t} = \frac{-32\pi}{60} = -\frac{32\pi}{60} = -\frac{8\pi}{15} \, \text{N m}
\]
The negative sign indicates that the torque is acting in the opposite direction of the rotation.
### Final Answer
The magnitude of the torque required to stop the wheel is:
\[
\tau = \frac{8\pi}{15} \, \text{N m}
\]
