The elevator E and its freight have a total mass of 400 kg....

The elevator $E$ and its freight have a total mass of 400 kg. Hoisting is provided by the motor $M$ and the 60 -kg block $C .$ If the motor has an efficiency of $\varepsilon=0.6,$ determine the power that must be supplied to the motor when the elevator is hoisted upward at a constant speed of $v_{E}=4 \mathrm{m} / \mathrm{s}.$

Key Concepts

Power

Power is the rate at which work is done or energy is transferred. In mechanical systems such as hoisting mechanisms, power is calculated by determining the work done against gravity (or any resisting force) per unit time. When an object is raised, the work done is the product of the weight of the object and the vertical distance moved, and dividing this work by the time taken gives the power output required.

Gravitational Potential Energy

Gravitational potential energy is the energy possessed by an object due to its position in a gravitational field. Raising an object, such as an elevator, increases this potential energy by an amount equal to its mass times the gravitational acceleration times the height change. This concept is crucial in hoisting problems where work is done to overcome gravity.

Efficiency

Efficiency in a mechanical system is the ratio of the useful output work to the total energy input. When a motor has an efficiency less than 100%, extra energy is required to compensate for losses such as heat, friction, and other inefficiencies. In hoisting applications, the actual power supplied must account for these losses by dividing the ideal power needed by the efficiency factor.

Constant Velocity Motion

Constant velocity motion indicates that an object is moving at a steady speed without accelerating. Under such conditions, the net force acting on the object is zero, meaning the force produced by the motor exactly balances the gravitational force. This simplifies the power calculation because it allows one to consider only the work done against gravity, without additional contributions from kinetic energy changes.

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