A 45 kW motor winds a cable around a drum while lifting a 2700 kg mass to a height of 6 m in 3 sec. The power lost to the friction (in kW) is: (take $g=10~m/s^{2})$

Question

Accepted Answer

To determine the power lost to friction, we first need to calculate the total power required to lift the mass and then compare it to the power supplied by the motor.

1. **Calculate the work done in lifting the mass:**

The work done (W) in lifting a mass can be calculated using the formula:
   $$
   W = m \cdot g \cdot h
   $$
   where:
   - $ m = 2700 \, \text{kg} $ (mass),
   - $ g = 10 \, \text{m/s}^2 $ (acceleration due to gravity),
   - $ h = 6 \, \text{m} $ (height).

Substituting the values, we get:
   $$
   W = 2700 \cdot 10 \cdot 6 = 162000 \, \text{Joules}
   $$

2. **Calculate the power required to lift the mass:**

Power (P) is the rate of doing work, given by:
   $$
   P = \frac{W}{t}
   $$
   where $ t = 3 \, \text{seconds} $.

Substituting the values, we get:
   $$
   P = \frac{162000}{3} = 54000 \, \text{Watts} = 54 \, \text{kW}
   $$

3. **Determine the power lost to friction:**

The motor provides 45 kW of power. However, the calculated power required to lift the mass is 54 kW. This indicates that there is a discrepancy due to power lost to friction.

The power lost to friction is the difference between the power required and the power supplied:
   $$
   \text{Power lost to friction} = 54 \, \text{kW} - 45 \, \text{kW} = 9 \, \text{kW}
   $$

Therefore, the power lost to friction is 9 kW, which corresponds to Option 4.

**Explanation of why other options are incorrect:**

- **Option 1 (5 kW):** This option is incorrect because the calculated power lost to friction is 9 kW, not 5 kW. The discrepancy between the required power (54 kW) and the supplied power (45 kW) is 9 kW.

- **Option 2 (0.5 kW):** This option is incorrect because it significantly underestimates the power lost to friction. The actual power lost is 9 kW, which is much higher than 0.5 kW.

- **Option 3 (0.8 kW):** This option is also incorrect as it underestimates the power lost to friction. The correct calculation shows a loss of 9 kW, not 0.8 kW.

In conclusion, the correct answer is Option 4, as the power lost to friction is indeed 9 kW.