Mechanical Power - Ben Borgers

Power measures the rate at which work is done and energy is transformed from one form to another. Power is how fast work is done.
- $\text{Power} = \frac{\text{Energy transformed}}{\text{time}}$
- $P = \frac{\Delta E}{\Delta t}$
- Measured in Watts: $1 W = \frac{1 \ \text{J}}{1 \ \text{S}}$
The watt is the SI unit for power, but power is sometimes also measured in horsepower.
- $1 \ \text{Hp (horsepower)} = 746 \ \text{W}$
- $1 \ \text{W} = 1 \ \text{J/s}$
Example problem: How much work must be done to lift a 350 kg piano 15 meters into the air in 5 minutes?
- Energy at the top is gravitational potential energy, $U_g = mgh = 350 \ \text{kg} \ (10 \ \text{m/s}^2) \ 15 \text{m} = 52,500 \ \text{J}$
- $P = \frac{52,500 \ \text{J}}{300 \ \text{s}} = 175 \ \text{W}$ $P = \frac{5 2 , 5 0 0 J}{3 0 0 s} = 175 W$
  - $= 0.23 \ \text{Hp}$ (using horsepower conversion from earlier)
Example problem: A 70 kg jogger runs up a flight of stairs in 4.0 s. The vertical height of the stairs is 4.5 m.
- Jogger’s output in watts: $P = \frac{\Delta U_g}{\Delta t} = \frac{\Delta mgh}{t} = \frac{(70)(10)(4.5)}{4.0} = 788 \ \text{W}$
- How much energy did this require? Energy is power multiplied by time ( $E = Pt$ $E = P t$ ), so $E = Pt = (788)(4.0) = 3152 \ \text{J}$ $E = P t = (788) (4.0) = 3152 J$
  - This is also just the value of $U_g = mgh$
Power is the rate at which work is done: $P = \frac{W}{t} = F \frac{\Delta x}{t}\cos(\theta) = F v\cos(\theta) = Fv$ $P = \frac{W}{t} = F \frac{Δ x}{t} cos (θ) = F v cos (θ) = F v$
- $v = \frac{\Delta x}{t} \ \text{(average velocity)}$
- $\cos(\theta)$ usually cancels out, because we’re going in a straight line
If a car generates 18 Hp when traveling at a steady 90 km/hr, what must be the average force exerted on the car due to friction and air resistance?
- $18 \ \text{Hp} = 13,428 \ \text{W}$
- $P = Fv$ , so $F = \frac{P}{v}$
- $F = \frac{13,428 \ \text{W}}{25 \ \text{m/s}} = 537 \ \text{N}$