The power of a crane, which lifts a mass of $1000 \text{ kg}$ to a height of $20 \text{ m}$ in $10 \text{ s}$ is :
($g = 9.8 \text{ m/s}^2$)
Calculate Crane Power
The problem asks for the power of a crane based on the mass it lifts, the height it lifts it to, and the time taken. We are given:
- Mass ($m$): $1000 \text{ kg}$
- Height ($h$): $20 \text{ m}$
- Time ($t$): $10 \text{ s}$
- Acceleration due to gravity ($g$): $9.8 \text{ m/s}^2$
Power is the rate at which work is done. First, we calculate the work done by the crane in lifting the mass against gravity.
Work Calculation
The work done ($W$) is equal to the potential energy gained by the mass, calculated using the formula:
$W = mgh$
Substituting the given values:
$W = (1000 \text{ kg}) \times (9.8 \text{ m/s}^2) \times (20 \text{ m})$
$W = 196000 \text{ J}$
Power Calculation
Power ($P$) is calculated by dividing the work done by the time taken:
$P = \frac{W}{t}$
Substituting the calculated work and given time:
$P = \frac{196000 \text{ J}}{10 \text{ s}}$
$P = 19600 \text{ W}$
Unit Conversion
The options are given in kilowatts (kW). To convert watts to kilowatts, we divide by 1000:
$P = \frac{19600 \text{ W}}{1000 \text{ W/kW}}$
$P = 19.6 \text{ kW}$
Therefore, the power of the crane is $19.6 \text{ kW}$.
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Important Questions from Mechanics
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