Question

Calculate the power of the crane in watts, which lifts a mass of \[100\,kg\] to a height of \[10m\] in \[20{\text{ }}seconds.\]

Answer 1

Hint: The term "power" refers to the ability to provide energy or force. Starting an engine is one example of power. The combination of forces and movement, in mechanical systems, is known as power. Here, power is defined as the product of a force on an object and the velocity of the object, or the product of torque on a shaft and the angular velocity of the shaft. Joules or watt-seconds are the units used to measure energy.

Complete step by step answer:
The Work-Energy theorem states that the work done on a system is equal to the change in energy of the system.The work done on the roller coaster car in lifting it to the top of the hill will be equal to the change in the gravitational potential energy of the car. Since \[GPE{\text{ }} = {\text{ }}mgh{\text{ }}\] and \[W = Fd = mgd\], thus the work done is equal to the gain in gravitational potential energy of the car.

The power is the rate of work, so,
\[P = \dfrac{W}{t}\]
By definition, work done is the product of applied force and displacement due to the force, so $W = Fd$
Where, $F$ is applied force and, $d$ is the displacement.
\[P = \dfrac{{Fd}}{t}\]
Here the applied force is equal to the weight of the body, so $F = mg$
\[P = \dfrac{{mgd}}{t}\]
\[ \Rightarrow P = \dfrac{{100 \times 10 \times 10}}{{20}}\]
\[ \therefore P = 500\,W\]

So, the power of the crane is \[500{\text{ }}watts\].

Note: The power expended can be negative if the force is antiparallel to the velocity or points in the opposite direction. The law of conservation of energy states that the total power of all elements in a circuit is zero since power given is a negative quantity and power absorbed is a positive quantity. There can never be more power absorbed than delivered or more power delivered than absorbed at any given time.