To calculate the input power of the motor, we first need to determine the useful power being delivered to the grinding wheel and then account for the efficiency of the machine.
Step 1: Calculate the Tangential Speed of the Grinding Wheel
The tangential speed \( V \) of the grinding wheel can be calculated using the formula:
\[ V = \pi \times D \times N \]
where:
- \( D \) is the diameter in meters (here, \( D = 200 \) mm = 0.2 m),
- \( N \) is the rotation speed in revolutions per minute (rpm).
First, calculate the perimeter \( C \) of the grinding wheel:
\[ C = \pi \times D = \pi \times 0.2 \text{ m} \approx 0.6283 \text{ m} \]
Next, convert the rotation speed from rpm to revolutions per second (rps):
\[ N = 3000 \text{ r/min} = \frac{3000}{60} \text{ r/s} = 50 \text{ r/s} \]
Now calculate \( V \):
\[ V = C \times N = 0.6283 \text{ m} \times 50 \text{ r/s} \approx 31.415 \text{ m/s} \]
Step 2: Calculate the Useful Power Delivered to the Grinding Wheel
The useful power \( P_{useful} \) can be calculated using the formula:
\[ P_{useful} = F \times V \]
where \( F \) is the tangential force.
Here, \( F = 40 \text{ N} \):
\[ P_{useful} = 40 \text{ N} \times 31.415 \text{ m/s} \approx 1256.6 \text{ W} \]
Step 3: Calculate the Input Power of the Motor
Given the efficiency of the machine \( \eta = 72% = 0.72 \), we can calculate the input power \( P_{input} \):
\[ \eta = \frac{P_{useful}}{P_{input}} \]
Rearranging this gives:
\[ P_{input} = \frac{P_{useful}}{\eta} \]
Now substituting the values of \( P_{useful} \) and \( \eta \):
\[ P_{input} = \frac{1256.6 \text{ W}}{0.72} \approx 1741.6 \text{ W} \]
Conclusion
The input power of the motor is approximately 1741.6 W, or about 1.74 kW.