8.2 Power

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Power

Now that we understand the relationship between work and energy, we are ready to look at a quantity that defines how long it takes for a certain amount of work to be done. For example, a mother pushing a trolley full of groceries can take 30 s or 60 s to push the trolley down an aisle. She does the same amount of work, but takes a different length of time. We use the idea of power to describe the rate at which work is done.

Power

Power is defined as the rate at which work is done or the rate at which energy is expended. The mathematical definition for power is:

$P=F·v$

[link] is easily derived from the definition of work. We know that:

$W=F·d.$

However, power is defined as the rate at which work is done. Therefore,

$P=\frac{\Delta W}{\Delta t}.$

This can be written as:

$\begin{array}{ccc}\hfill P& =& \frac{\Delta W}{\Delta t}\hfill \\ & =& \frac{\Delta \left(F·d\right)}{\Delta t}\hfill \\ & =& F\frac{\Delta d}{\Delta t}\hfill \\ & =& F·v\hfill \end{array}$

The unit of power is watt (symbol W).

Investigation : watt

Show that the W is equivalent to $\mathrm{J}·{\mathrm{s}}^{-1}$ .

Interesting fact

The unit watt is named after Scottish inventor and engineer James Watt (19 January 1736 - 19 August 1819) whose improvements to the steam engine were fundamental to the Industrial Revolution. A key feature of it was that it brought the engine out of the remote coal fields into factories.

Research project : james watt

Write a short report 5 pages on the life of James Watt describing his many other inventions.

Interesting fact

Historically, the horsepower (symbol hp) was the unit used to describe the power delivered by a machine. One horsepower is equivalent to approximately 750 W. The horsepower is sometimes used in the motor industry to describe the power output of an engine. Incidentally, the horsepower was derived by James Watt to give an indication of the power of his steam engine in terms of the power of a horse, which was what most people used to for example, turn a mill wheel.

Calculate the power required for a force of 10 N applied to move a 10 kg box at a speed of 1 ms over a frictionless surface.

1. We are given:

• we are given the force, $F$ =10 N
• we are given the speed, $v$ =1 m $·$ s ${}^{-1}$

We are required to calculate the power required.

2. From the force diagram, we see that the weight of the box is acting at right angles to the direction of motion. The weight does not contribute to the work done and does not contribute to the power calculation.

We can therefore calculate power from:

$P=F·v$
3. $\begin{array}{ccc}\hfill P& =& F·v\hfill \\ & =& \left(10\phantom{\rule{0.166667em}{0ex}}\mathrm{N}\right)\left(1\phantom{\rule{0.166667em}{0ex}}\mathrm{m}·{\mathrm{s}}^{-1}\right)\hfill \\ & =& 10\phantom{\rule{0.166667em}{0ex}}\mathrm{W}\hfill \end{array}$
4. 10 W of power are required for a force of 10 N to move a 10 kg box at a speed of 1 ms over a frictionless surface.

Machines are designed and built to do work on objects. All machines usually have a power rating. The power rating indicates the rate at which that machine can do work upon other objects.

A car engine is an example of a machine which is given a power rating. The power rating relates to how rapidly the car can accelerate. Suppose that a 50 kW engine could accelerate the car from 0  $\mathrm{km}·{\mathrm{hr}}^{-1}$ to 60 $\mathrm{km}·{\mathrm{hr}}^{-1}$ in 16 s. Then a car with four times the power rating (i.e. 200 kW) could do the same amount of work in a quarter of the time. That is, a 200 kW engine could accelerate the same car from 0  $\mathrm{km}·{\mathrm{hr}}^{-1}$ to 60 $\mathrm{km}·{\mathrm{hr}}^{-1}$ in 4 s.

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