# 8.2 Power

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A forklift lifts a crate of mass 100 kg at a constant velocity to a height of 8 m over a time of 4 s. The forklift then holds the crate in place for 20 s. Calculate how much power the forklift exerts in lifting the crate? How much power does the forklift exert in holding the crate in place?

1. We are given:

• mass of crate: $m$ =100 kg
• height that crate is raised: $h$ =8 m
• time taken to raise crate: ${t}_{r}$ =4 s
• time that crate is held in place: ${t}_{s}$ =20 s

We are required to calculate the power exerted.

2. We can use:

$P=F\frac{\Delta x}{\Delta t}$

to calculate power. The force required to raise the crate is equal to the weight of the crate.

3. $\begin{array}{ccc}\hfill P& =& F\frac{\Delta x}{\Delta t}\hfill \\ & =& m·g\frac{\Delta x}{\Delta t}\hfill \\ & =& \left(100\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}\right)\left(9,8\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-2}\right)\frac{8\phantom{\rule{0.166667em}{0ex}}\mathrm{m}}{4\phantom{\rule{0.166667em}{0ex}}\mathrm{s}}\hfill \\ & =& 1\phantom{\rule{0.277778em}{0ex}}960\phantom{\rule{0.166667em}{0ex}}\mathrm{W}\hfill \end{array}$
4. While the crate is being held in place, there is no displacement. This means there is no work done on the crate and therefore there is no power exerted.

5. 1 960 W of power is exerted to raise the crate and no power is exerted to hold the crate in place.

## Experiment : simple measurements of human power

You can perform various physical activities, for example lifting measured weights or climbing a flight of stairs to estimate your output power, using a stop watch. Note: the human body is not very efficient in these activities, so your actual power will be much greater than estimated here.

## Power

1. [IEB 2005/11 HG] Which of the following is equivalent to the SI unit of power:
1. V $·$ A
2. V $·$ A ${}^{-1}$
3. $kg·m·{s}^{-1}$
4. kg $·$ m $·{s}^{-2}$
2. Two students, Bill and Bob, are in the weight lifting room of their local gum. Bill lifts the 50 kg barbell over his head 10 times in one minute while Bob lifts the 50 kg barbell over his head 10 times in 10 seconds. Who does the most work? Who delivers the most power? Explain your answers.
3. Jack and Jill ran up the hill. Jack is twice as massive as Jill; yet Jill ascended the same distance in half the time. Who did the most work? Who delivered the most power? Explain your answers.
4. Alex (mass 60 kg) is training for the Comrades Marathon. Part of Alex's training schedule involves push-ups. Alex does his push-ups by applying a force to elevate his center-of-mass by 20 cm. Determine the number of push-ups that Alex must do in order to do 10 J of work. If Alex does all this work in 60 s, then determine Alex's power.
5. When doing a chin-up, a physics student lifts her 40 kg body a distance of 0.25 m in 2 s. What is the power delivered by the student's biceps?
6. The unit of power that is used on a monthly electricity account is kilowatt-hours (symbol kWh). This is a unit of energy delivered by the flow of l kW of electricity for 1 hour. Show how many joules of energy you get when you buy 1 kWh of electricity.
7. An escalator is used to move 20 passengers every minute from the first floor of a shopping mall to the second. The second floor is located 5-meters above the first floor. The average passenger's mass is 70 kg. Determine the power requirement of the escalator in order to move this number of passengers in this amount of time.
8. Calculate the power required for an electric motor to pump 20kg of water up to ground level from a borehole of depth 10m in half a minute.

These two videos provide a summary of some of the concepts covered in this chapter.

## Important equations and quantities

 Units Quantity Symbol Unit S.I. Units Direction velocity $\stackrel{\to }{v}$ — $\frac{m}{s}$ or $\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}m.{s}^{-1}$ momentum $\stackrel{\to }{p}$ — $\frac{kg.m}{s}$ or $\phantom{\rule{4pt}{0ex}}kg.m.{s}^{-1}$ energy $E$ $J$ $\frac{kg.{m}^{2}}{{s}^{2}}$ or $\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}.{m}^{2}{s}^{-2}$ — Work $W$ J $N.m$ or $kg.{m}^{2}.{s}^{-2}$ — Kinetic Energy ${E}_{K}$ J $N.m$ or $kg.{m}^{2}.{s}^{-2}$ — Potential Energy ${E}_{P}$ J $N.m$ or $kg.{m}^{2}.{s}^{-2}$ — Mechanical Energy $U$ J $N.m$ or $kg.{m}^{2}.{s}^{-2}$ — Power P W $N.m.{s}^{-1}$ or $kg.{m}^{2}.{s}^{-3}$ —

Momentum :

$\stackrel{\to }{p}=m\stackrel{\to }{v}$

Kinetic energy :

${E}_{k}=\frac{1}{2}m{\stackrel{\to }{v}}^{2}$
• Energy is never created nor destroyed, but is merely transformed from one form to another.
• In the absence of friction, the total mechanical energy of an object is conserved.

When a force moves in the direction along which it acts, work is done.

Work is the process of converting energy.

Energy is the ability to do work.

## End of chapter exercises

1. The force vs. displacement graph shows the amount of force applied to an object by three different people. Abdul applies force to the object for the first 4 m of its displacement, Beth applies force from the 4 m point to the 6 m point, and Charles applies force from the 6 m point to the 8 m point. Calculate the work done by each person on the object? Which of the three does the most work on the object?
2. How much work does a person do in pushing a shopping trolley with a force of 200 N over a distance of 80 m in the direction of the force?
3. How much work does the force of gravity do in pulling a 20 kg box down a 45 ${}^{\circ }$ frictionless inclined plane of length 18 m?
4. [IEB 2001/11 HG1] Of which one of the following quantities is kg.m ${}^{2}$ .s ${}^{-3}$ the base S.I. unit?
1. Energy
2. Force
3. Power
4. Momentum
5. [IEB 2003/11 HG1] A motor is used to raise a mass m through a vertical height h in time t.What is the power of the motor while doing this?
1. $mght$
2. $\frac{mgh}{t}$
3. $\frac{mgt}{h}$
4. $\frac{ht}{mg}$
6. [IEB 2002/11 HG1] An electric motor lifts a load of mass M vertically through a height h at a constant speed v. Which of the following expressions can be used to correctly calculate the power transferred by the motor to the load while it is lifted at a constant speed?
1. $Mgh$
2. $Mgh+\frac{1}{2}$ Mv ${}^{2}$
3. $Mgv$
4. $Mgv+\frac{1}{2}$ $\frac{{\mathrm{Mv}}^{3}}{\mathrm{h}}$
7. [IEB 2001/11 HG1] An escalator is a moving staircase that is powered by an electric motor. People are lifted up the escalator at a constant speed of v through a vertical height h.What is the energy gained by a person of mass m standing on the escalator when he is lifted from the bottom to the top?
1. mgh
2. mgh $sin\theta$
3. $\frac{\mathrm{mgh}}{sin\theta }$
4. $\frac{1}{2}$ mv ${}^{2}$
8. [IEB 2003/11 HG1] In which of the following situations is there no work done on the object?
1. An apple falls to the ground.
2. A brick is lifted from the ground to the top of a building.
3. A car slows down to a stop.
4. A box moves at constant velocity across a frictionless horizontal surface.

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