Question

The power of a crane, which lifts a mass of 1000 kg to a height of 20 m in 10 s is: (g = 9.8 m/s²)

• A. 39.2 kW
• B. 39.2 W
• C. 19.6 kW
• D. 19.6 W

Hint:

Always double-check the units in the options. 19.6 W and 19.6 kW are both present as distractors; ensure you convert Watts to kilowatts correctly by dividing by 1000.

Answer 1

The Correct Option is C

Step 1: Understanding the Topic:
This problem relates to "Work, Energy, and Power." Power is defined as the rate at which work is performed or energy is transferred. In this scenario, the crane is doing work against the force of gravity to increase the gravitational potential energy of the mass.
Step 2: Key Formulas and Approach:

Work Done ($W$) against gravity = $mgh$.
Power ($P$) = $\text{Work Done } / \text{Time taken}$.
Units: Work is in Joules (J), Power is in Watts (W), and $1000 \text{ W} = 1 \text{ kW}$.

Step 3: Detailed Explanation:

Identify given values: Mass $m = 1000 \text{ kg}$, Height $h = 20 \text{ m}$, Time $t = 10 \text{ s}$, and $g = 9.8 \text{ m/s}^2$.
Calculate Work Done: The crane must exert a force equal to the weight of the object over the given height. \[ W = m \times g \times h = 1000 \text{ kg} \times 9.8 \text{ m/s}^2 \times 20 \text{ m} \] \[ W = 196,000 \text{ Joules (or 196 kJ)} \]
Calculate Power: Power is the work divided by the duration of the lift. \[ P = \frac{W}{t} = \frac{196,000 \text{ J}}{10 \text{ s}} = 19,600 \text{ Watts} \]
Convert to kilowatts: Since the options are in kW, we divide the result by 1000. \[ P = \frac{19,600}{1000} = 19.6 \text{ kW} \]
This result represents the average power output of the crane during the lifting process.
Step 4: Final Answer:
The power of the crane is 19.6 kW.