An automobile engine develops `100` kilo`-`watt, when rotating at a speed of `1800 rev//min`. Find the torque developed by it.

Options

A. `10^(2)/6pi` N m
B. `10^(4)/6pi` N m
C. `10^(6)/6pi` N m
D. `10^(8)/6pi` N m

Explanation

To find the torque developed by the automobile engine, we can follow these steps:

### Step 1: Identify the Given Values
- Power (P) = 100 kilowatts = \(100 \times 10^3\) watts
- Speed (N) = 1800 revolutions per minute (rev/min)

### Step 2: Convert the Speed from Revolutions per Minute to Radians per Second
To convert revolutions per minute to radians per second, we use the conversion factor:
\[ \text{Angular speed} (\omega) = N \times \frac{2\pi \text{ radians}}{1 \text{ revolution}} \times \frac{1 \text{ minute}}{60 \text{ seconds}} \]
Substituting the values:
\[ \omega = 1800 \times \frac{2\pi}{60} \]
Calculating this gives:
\[ \omega = 1800 \times \frac{2\pi}{60} = 1800 \times \frac{\pi}{30} = 60\pi \text{ radians/second} \]

### Step 3: Use the Formula for Torque
The formula for torque (\(\tau\)) in terms of power and angular speed is given by:
\[ \tau = \frac{P}{\omega} \]
Substituting the values we have:
\[ \tau = \frac{100 \times 10^3}{60\pi} \]

### Step 4: Calculate the Torque
Now, we can calculate the torque:
\[ \tau = \frac{100000}{60\pi} \text{ Newton meter} \]
This simplifies to:
\[ \tau \approx \frac{100000}{188.4} \approx 530.5 \text{ Newton meter} \]

### Final Answer
Thus, the torque developed by the automobile engine is approximately:
\[ \tau \approx \frac{100000}{60\pi} \text{ Newton meter} \]

---