An automobiles engine develops a power of 100 kilowatt, when rotating at a speed of 30 rev/sec. What torque does it deliver?

Options

A. `(1000)/(3pi)N-m`
B. `(2000)/(3pi)N-m`
C. `(5000)/(3pi)N-m`
D. `(4000)/(3pi)N-m`

Explanation

To find the torque delivered by the automobile engine, we can use the relationship between power, torque, and angular velocity. The steps to solve the problem are as follows:

### Step 1: Convert Power to Watts
The power of the engine is given as 100 kilowatts. We need to convert this into watts:
\[ \text{Power} (P) = 100 \text{ kW} = 100 \times 10^3 \text{ W} = 100,000 \text{ W} \]

### Step 2: Convert Rotational Speed to Radians per Second
The rotational speed is given as 30 revolutions per second. We need to convert this to radians per second since the formula for power in terms of torque and angular velocity uses radians:
\[ \text{Angular velocity} (\omega) = 30 \text{ rev/sec} \times 2\pi \text{ rad/rev} = 60\pi \text{ rad/sec} \]

### Step 3: Use the Power-Torque Relationship
The relationship between power (P), torque (τ), and angular velocity (ω) is given by:
\[ P = \tau \cdot \omega \]
We can rearrange this equation to solve for torque (τ):
\[ \tau = \frac{P}{\omega} \]

### Step 4: Substitute the Values
Now, we can substitute the values of power and angular velocity into the equation:
\[ \tau = \frac{100,000 \text{ W}}{60\pi \text{ rad/sec}} \]

### Step 5: Calculate Torque
Now we will calculate the torque:
\[ \tau = \frac{100,000}{60\pi} \approx \frac{100,000}{188.4} \approx 530.5 \text{ Nm} \]

### Final Answer
The torque delivered by the engine is approximately:
\[ \tau \approx 530.5 \text{ Nm} \]

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