How much work does an elevator motor do to lift a 1000 kg elevator a height of 100 m? How much power must the motor supply to do this in 50 s at constant speed?

Verified step by step guidance

1. Step 1: Identify the given values in the problem. The mass of the elevator is \( m = 1000 \; \text{kg} \), the height it is lifted is \( h = 100 \; \text{m} \), the time taken is \( t = 50 \; \text{s} \), and the acceleration due to gravity is \( g = 9.8 \; \text{m/s}^2 \).
2. Step 2: Calculate the work done by the motor to lift the elevator. Work is given by the formula \( W = m g h \), where \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the height. Substitute the given values into the formula to find \( W \).
3. Step 3: To calculate the power supplied by the motor, use the formula \( P = \frac{W}{t} \), where \( W \) is the work done and \( t \) is the time. Substitute the value of \( W \) from Step 2 and the given \( t \) into this formula.
4. Step 4: Note that the elevator is moving at constant speed, so there is no change in kinetic energy. This means the work done by the motor is entirely used to overcome the gravitational force and lift the elevator.
5. Step 5: Ensure that the units of the final results for work (in joules) and power (in watts) are consistent with the SI system. This step ensures the calculations are dimensionally correct.

Key Concepts

Work

Work is defined as the product of force and displacement in the direction of the force. In the context of lifting an elevator, the work done by the motor can be calculated using the formula W = F × d, where W is work, F is the force (equal to the weight of the elevator), and d is the height lifted. The weight can be determined by multiplying the mass of the elevator by the acceleration due to gravity (approximately 9.81 m/s²).

Power

Power is the rate at which work is done or energy is transferred over time. It can be calculated using the formula P = W/t, where P is power, W is work, and t is the time taken to do the work. In this scenario, once the work done to lift the elevator is determined, the power required by the motor to lift it in a specified time can be calculated, providing insight into the motor's performance.

Constant Speed

When an object moves at constant speed, it means that its velocity remains unchanged over time, implying that the net force acting on it is zero. In the case of the elevator, lifting it at constant speed means that the upward force exerted by the motor equals the downward gravitational force acting on the elevator. This condition simplifies the calculations for work and power, as the motor's force directly counteracts the weight of the elevator.