Power
Power
Now that we understand the relationship between work and energy, we are ready to look at a quantity related the rate of energytransfer. For example, a mother pushing a trolley full of groceries can take \(\text{30}\) \(\text{s}\) or \(\text{60}\) \(\text{s}\) 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.
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.
Definition: Power
Power is defined as the rate at which work is done or the rate at which energy is transfered to or from a system. The mathematical definition for power is:
\(P=\frac{W}{t}\)Power is easily derived from the definition of work. We know that:
\(W=F \Delta x \cos \theta\)
Power is defined as the rate at which work is done. Therefore,
\begin{align*} P & = \frac{W}{t} \\ & = \frac{F \Delta x \cos \theta}{t} \\ & \text{in the case where }F \text{ and }\Delta x \text{ are in the same direction} \\ & = \frac{F \Delta x}{t} \\ & = F \frac{ \Delta x}{t} \\ & = F v \end{align*}
Important:
In the case where the force and the velocity are in opposite directions the power will be negative.
The unit of power is watt (symbol W).
Fact:
Historically, the horsepower (symbol hp) was the unit used to describe the power delivered by a machine. One horsepower is equivalent to approximately \(\text{750}\) \(\text{W}\). 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.
Example: Power Calculation
Question
Calculate the power required for a force of \(\text{10}\) \(\text{N}\) applied to move a \(\text{10}\) \(\text{kg}\) box at a speed of \(\text{1}\) \(\text{m·s$^{-1}$}\) over a frictionless surface.
Solution
Step 1: Determine what is given and what is required.
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We are given the force, \(F=10~\text{N}\).
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We are given the speed, \(v=1~\text{m}·{\text{s}}^{-1}\).
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We are required to calculate the power required.
Step 2: Draw a force diagram
Step 3: Determine how to approach the problem
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: \(P=F·v\).
Step 4: Calculate the power required
\begin{align*} P& = F·v\\ & = \left(10 \text{N}\right)\left(1 \text{m}·{\text{s}}^{-1}\right)\\ & = 10 \text{W} \end{align*}Step 5: Write the final answer
\(\text{10}\) \(\text{W}\) of power are required for a force of \(\text{10}\) \(\text{N}\) to move a \(\text{10}\) \(\text{kg}\) box at a speed of \(\text{1}\) \(\text{m·s$^{-1}$}\) 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 \(\text{0}\) \(\text{km·hr$^{-1}$}\) to \(\text{60}\) \(\text{km·hr$^{-1}$}\) in \(\text{16}\) \(\text{s}\). Then a car with four times the power rating (i.e. \(\text{200}\) \(\text{kW}\)) could do the same amount of work in a quarter of the time. That is, a \(\text{200}\) \(\text{kW}\) engine could accelerate the same car from \(\text{0}\) \(\text{km·hr$^{-1}$}\) to \(\text{60}\) \(\text{km·hr$^{-1}$}\) in \(\text{4}\) \(\text{s}\).
This lesson is part of:
Work, Energy and Power