A motor generates an output power of 220 W at an angular velocity of 2100 rpm. Calculate the torque ( in Nm ) produced by the motor ? [ Take `pi= ( 22)/( 7) `]

Answer

To solve the problem of calculating the torque produced by a motor given its output power and angular velocity, we can follow these steps:

### Step 1: Understand the Given Values
- Power (P) = 220 W
- Angular velocity (ω) = 2100 rpm

### Step 2: Convert Angular Velocity to Radians per Second
To convert the angular velocity from revolutions per minute (rpm) to radians per second (rad/s), we use the conversion factor:
\[ \text{Angular velocity in rad/s} = \text{Angular velocity in rpm} \times \frac{2\pi \text{ rad}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}} \]
Substituting the given value:
\[ \omega = 2100 \times \frac{2 \times \frac{22}{7}}{60} \]
Calculating this gives:
\[ \omega = 2100 \times \frac{44}{420} = 2100 \times \frac{1}{9.545} \approx 70 \times \frac{22}{7} \text{ rad/s} \]
Thus,
\[ \omega \approx 70 \times \frac{22}{7} \text{ rad/s} \]

### Step 3: Use the Power-Torque Relationship
The relationship between power (P), torque (τ), and angular velocity (ω) is given by:
\[ P = τ \cdot ω \]
From this, we can express torque as:
\[ τ = \frac{P}{ω} \]

### Step 4: Substitute the Values
Substituting the values we have:
\[ τ = \frac{220 \text{ W}}{70 \times \frac{22}{7} \text{ rad/s}} \]

### Step 5: Simplify the Expression
Now, simplifying the expression:
\[ τ = \frac{220}{70 \times \frac{22}{7}} = \frac{220 \times 7}{70 \times 22} \]
The 70 and 220 can be simplified:
\[ = \frac{31.42857}{22} \approx 1 \text{ Nm} \]

### Final Answer
Thus, the torque produced by the motor is approximately:
\[ τ \approx 1 \text{ Nm} \]