Power in Mechanics Problems

1. Elevator Lifted at Constant Speed

Even if the elevator isn’t accelerating, power is still required to lift it against gravity. Since it’s moving at a constant velocity, the net force is zero, but the motor must apply an upward force equal to the weight.

\( P = F \cdot v = mgv \)

\( m = 600\,\text{kg},\ g = 9.8\,\text{m/s}^2,\ v = 2\,\text{m/s} \)

\( P = 600 \cdot 9.8 \cdot 2 = 11,\!760\,\text{W} \)

2. Car Accelerating with Engine Force

As a car speeds up, its velocity increases. If the engine force remains constant, power increases with velocity.

\( P(t) = F \cdot v(t) \)

Example: \( F = 2000\,\text{N},\ v(t) = 3t \Rightarrow P(t) = 2000 \cdot 3t = 6000t\,\text{W} \)

3. Graphical Interpretation of Power

Power vs Time: Area under the curve = Energy
Work vs Time: Slope = Power
Force vs Velocity: Area under the curve = Power delivered