A crane lifts a mass m to the 10th floor of a building in a certain time. It then raises the same mass to the 20th floor of the same building in double the time. How much more energy and power are required? Assume that the height of all floors is equal.
Let the height of one floor be h. For the 10th floor, height is 10h and time is t. For the 20th floor, height is 20h and time is 2t.
Energy required: Energy for 10th floor, E1 = mg × 10h = 10mgh. Energy for 20th floor, E2 = mg × 20h = 20mgh. Since E2 = 2E1, double the energy is required.
Power required: Power for 10th floor, P1 = E1 / t = 10mgh / t. Power for 20th floor, P2 = E2 / 2t = 20mgh / 2t = 10mgh / t. Since P2 = P1, no more power is required; the power remains the same.
Explanation
The context establishes that work done (energy) depends on the force and the distance (height) over which it acts. Lifting the mass to the 20th floor covers double the height compared to the 10th floor, thus requiring double the energy. Power is defined as work done per unit time. Since the time taken is also doubled for the 20th floor, the ratio of work to time (power) remains exactly the same as for the 10th floor.
Solution Steps
Step 1: Let height of one floor be h. For 10th floor, height = 10h, time = t. For 20th floor, height = 20h, time = 2t.
Step 2: Calculate Energy: E1 = mg × 10h = 10mgh; E2 = mg × 20h = 20mgh. E2 is double E1, so double the energy is required.
Step 3: Calculate Power: P1 = E1 / t = 10mgh / t; P2 = E2 / 2t = 20mgh / 2t = 10mgh / t. P2 = P1, so no more power is required.