Question

A car of mass 2000 kg is lifted up a distance of 30 m by a crane in 1 minute. A second crane does the same job in 2 minutes. Do the cranes consume the same or different amounts
of fuel? What is the power supplied by each crane? Neglect power dissipation against friction.
Take g = 10 ms^–2.

Answer 1

Step 1: Given data

Mass of the car is, $m=2000kg$

Height through which the car is lifted is, $h=30m$

Time taken by the first crane to lift the car is, $t_2=1min=60s$

Time taken by the second crane to lift the car is, $t_2=2min=120s$

Acceleration due to gravity is to be taken as, $g=10m/s^2$

Step 2: Calculating work done

Work done is given as,

$W=mgh$

where, $m$ is mass, $g$ is the acceleration due to gravity and $h$ is height.

Substituting the known values,

$$\begin{gathered} W=2000kg\times 10m/s^2\times 30m \\ W=6\times {10}^{5}J \end{gathered}$$

Since work done does not depend upon time, both cranes do the same amount of work and therefore, use the same amount of fuel.

Step 3: Calculating power

Power is given as,

$P=\frac{W}{t}$

where, $W$ is work done and $t$ is time.

For the first crane, power can be calculated as,

$$\begin{gathered} P=\frac{W}{t_1} \\ P=\frac{6\times {10}^{5}J}{60s} \\ P={10}^{4}W \end{gathered}$$

For the second crane, power can be calculated as,

$$\begin{gathered} P=\frac{W}{t_2} \\ P=\frac{6\times {10}^{5}J}{120s} \\ P=5\times {10}^{3}W \end{gathered}$$

Therefore, the cranes consume the same amounts of fuel. The power supplied by the first crane is ${10}^{4}W$ and the power supplied by the second crane is $5\times {10}^{3}W$.