A wheel is rotating about an axis through its centre at 720 rpm . It is acted on by a constant torque opposing its motion for 8 seconds to bring it to rest finally . The value of torque ( in N-m ) Is :- (Given `I = (24)/pi kg - m^(2)`)

Options

A. 48
B. 72
C. 96
D. 120

Explanation

To solve the problem step by step, we will follow the given information and apply the relevant physics concepts.

### Step 1: Convert Angular Velocity from RPM to Radians per Second
The initial angular velocity (ω_i) is given as 720 revolutions per minute (rpm). To convert this to radians per second, we use the conversion factor:
\[ \text{1 revolution} = 2\pi \text{ radians} \]
\[ \text{1 minute} = 60 \text{ seconds} \]
Thus, we can convert 720 rpm to radians per second:
\[ \omega_i = 720 \, \text{rpm} \times \frac{2\pi \, \text{radians}}{1 \, \text{revolution}} \times \frac{1 \, \text{minute}}{60 \, \text{seconds}} = 720 \times \frac{2\pi}{60} \]
Calculating this gives:
\[ \omega_i = 720 \times \frac{2\pi}{60} = 720 \times \frac{1}{30} \times 2\pi = 24\pi \, \text{radians/second} \]

### Step 2: Identify Final Angular Velocity
The final angular velocity (ω_f) when the wheel comes to rest is:
\[ \omega_f = 0 \, \text{radians/second} \]

### Step 3: Use the Angular Motion Equation
We can use the angular motion equation:
\[ \omega_f = \omega_i + \alpha t \]
Where:
- \( \alpha \) is the angular acceleration,
- \( t \) is the time duration (8 seconds).

Substituting the known values:
\[ 0 = 24\pi + \alpha \cdot 8 \]
Rearranging to find α:
\[ \alpha \cdot 8 = -24\pi \]
\[ \alpha = -\frac{24\pi}{8} = -3\pi \, \text{radians/second}^2 \]

### Step 4: Calculate Torque
The torque (τ) can be calculated using the formula:
\[ \tau = I \cdot \alpha \]
Where:
- \( I \) is the moment of inertia, given as \( \frac{24}{\pi} \, \text{kg m}^2 \).

Substituting the values:
\[ \tau = \left(\frac{24}{\pi}\right) \cdot (-3\pi) \]
Simplifying this:
\[ \tau = -\frac{24 \cdot 3\pi}{\pi} = -72 \, \text{N-m} \]
Since we are interested in the magnitude of torque, we take the positive value:
\[ \tau = 72 \, \text{N-m} \]

### Final Answer
The value of the torque is \( 72 \, \text{N-m} \).

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