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Work, Kinetic Energy and Power

Why this chapter

Why this chapter exists

Newton's laws tell you the acceleration of a body, and from it, step by step, its speed. This chapter gives a shorter road. A force that moves its point of application transfers energy, and that transfer is called work. The net work of all the forces on a body equals the change in its energy of motion, its kinetic energy. So a speed can be found from forces and distances alone, with no acceleration and no time. The same idea explains why a car at twice the speed needs four times the road to stop, and why a hill needs more power. It is also the first half of energy conservation, the subject of the next chapter.

You will learn

By the end of this chapter you will

Find the work of a constant force, with its sign, as \(Fd\cos\theta\) or as a scalar product, and recognise the two kinds of zero work.
Find the work of a variable force as the signed area under its \(F\)-\(x\) graph or as an integral, and a spring's \(\tfrac12 kx^2\).
State the work of gravity, the normal force, kinetic and static friction and tension, and the conditions under which each is zero.
Explain when a pair of internal forces does net work, and where that energy comes from or goes.
Use \(K = \tfrac12 mv^2\) and the work-energy theorem, with the net work of every force, to find speeds and stopping distances.
Use \(P = W/\Delta t\), \(P = dW/dt\) and \(P = \vec F\cdot\vec v\) for engines on the level and on slopes, and for pumps.

Before you begin

You should know Newton's three laws and how to draw a free-body diagram. You should know the weight, the normal force, tension, the spring force \(kx\), and static and kinetic friction. You should know the equations of motion for constant acceleration and the pseudo force in an accelerating frame. You should be able to resolve a vector into components and integrate a power of \(x\).

Already know this?

Lessons

Work Done by a Constant Force

Work is the energy a force transfers to a body as the body moves. For a constant force \(W = F \cdot d = Fd\cos\theta\), measured in joules, and its sign comes from the angle alone.

Work transfers energyThe joule and its sizeSign of work from the angle

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Work Done by a Variable Force

When a force changes as the body moves, the work is the area under the F-x graph, \(W = \int F\,dx\), and area below the axis counts as negative. Along a curved path it is the line integral of \(F \cdot dr\). Stretching a spring by \(x\) takes \(\tfrac12 kx^2\).

Work is area under the f x graphWork as an integralWork to stretch a spring

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Work Done by the Common Forces

Each everyday force asked the same question. Gravity does \(-mg\Delta h\) whatever the path, the normal force usually does none, kinetic friction does negative work, static friction can do positive work, and a string over a fixed pulley does no net work.

Work done by gravityWork done by the normal forceWork done by frictionWork done by tension

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Work Done by Internal Forces

The internal forces of a system cancel as vectors, but their works cancel only when their points of application move equally. When they do not, a store of energy inside the system is spent, or friction turns motion into heat.

Internal forces cancel as vectorsInternal work need not cancelInternal work spends a store

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Kinetic Energy and the Work-Energy Theorem

A moving body carries kinetic energy \(K = \tfrac12 mv^2\). The net work of all the forces on it equals the change in \(K\), which turns forces and distances straight into speeds, and makes the stopping distance grow as the square of the speed.

Kinetic energyWork energy theoremNet work is the sum of all worksStopping distance and speed

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Power

Power is the rate of doing work, \(P = W/\Delta t\) on average and \(dW/dt\) at an instant, and \(P = F \cdot v\) for a force on a moving body. It explains why a hill costs more power, what an engine does at a steady speed, and how a pump is rated.

Average and instantaneous powerPower is force times velocityPower at steady speedPower of a pump

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Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.

The Workbook

Questions to solve on paper, marked by you.

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Glossary

The key words of this chapter, with the meaning this chapter gives them.

efficiency: The ratio of the useful power a machine gives out to the power it takes in, \(\eta = P_{\text{out}}/P_{\text{in}}\). It is always less than one.
external force: A force on a member of a system from a body outside the system.
horsepower: An older unit of power, used for engines: \(1\,\text{hp} \approx 746\,\text{W}\).
internal force: A force that one member of a system exerts on another member. Internal forces come in third-law pairs, so their vector sum is zero.
joule: The SI unit of work and energy. One joule is the work done when a force of one newton moves its point of application one metre along its own direction: \(1\,\text{J} = 1\,\text{N m}\).
kilowatt hour: The work done in one hour at a power of one kilowatt, \(3.6 \times 10^6\,\text{J}\). It is the "unit" of an electricity bill.
kinetic energy: The energy a body has because it moves, \(K = \tfrac12 mv^2\). It is a scalar and is never negative.
line integral: The work of a force along a path, found by adding \(\vec F\cdot d\vec r\) over every tiny step of the path: \(\int_A^B \vec F\cdot d\vec r\).
net work: The sum of the works done by all the forces acting on a body, each with its sign.
point of application: The point of a body at which a force acts. The work of a force depends on how this point moves.
power: The rate at which work is done or energy is transferred, \(P = dW/dt\). For a force on a moving body, \(P = \vec F\cdot\vec v\). It is measured in watts.
scalar product: The product of two vectors that gives a number: \(\vec A\cdot\vec B = AB\cos\theta = A_xB_x + A_yB_y + A_zB_z\). It is also called the dot product.
stopping distance: The distance a moving body covers after a steady stopping force starts to act, \(s = mv^2/2F\). It grows as the square of the speed.
system: A group of bodies chosen for study. Everything else is outside the system.
watt: The SI unit of power: one joule per second, \(1\,\text{W} = 1\,\text{J/s}\).
work: The energy transferred to or from a body by a force as its point of application moves. For a constant force \(W = \vec F\cdot\vec d = Fd\cos\theta\). It is a scalar, measured in joules.
work-energy theorem: The net work done on a body by all the forces on it equals the change in its kinetic energy: \(W_{\text{net}} = K_f - K_i\).