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3.6.1Forces and Motion

Work Done

5 min read 8 parts Lesson 21 of 66 1 quiz Virtual lab

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Lesson summary

3.6.1 Work Done — OCR A Level Physics A notes: Define work done by a force, including when the force is at an angle, and interpret force–displacement graphs.Work done by a force is the energy transferred when the force moves its point of application.

  • area under the graph
  • Worked example 1
  • Worked example 2
  • Worked example 3
  • Component of force
  • Negative work

Section 1 of 8Definition

Work done by a force is the energy transferred when the force moves its point of application.

Definition
For a constant force F moving an object a distance s in the direction of the force, W = Fs, where W is in joules (J), F in newtons and s in metres.
One joule is the work done when a force of one newton moves an object one metre in the direction of the force.
Work is a scalar quantity.

Section 2 of 8Force at an angle

Only the component of force along the displacement does work.

What the symbols mean:
If the force acts at angle θ to the direction of motion, W = Fs cos θ.
What the symbols mean: W = work done = energy transferred (J) · F = force (N) · d, s or x = distance moved in the direction of the force (m)
For θ = 0° the full force counts; for θ = 90° (for example, the normal force on a body sliding along a horizontal surface,
  • or the tension supplying circular motion) no work is done.
If the force opposes the motion (θ = 180°), cos θ = −1 and the work done by the force is negative, meaning the object loses energy.
Friction therefore does negative work on a sliding block, transferring its kinetic energy to thermal energy.

Lifting a mass m at constant speed through height h needs a force equal to the weight mg, so the work done against gravity is mgh. Pulling a sledge with a rope at 30° above the horizontal through 50 m with a 60 N tension does 60 × 50 × cos 30° joules, not 60 × 50.

Section 3 of 8Variable forces and graphs

When the force varies, we plot force against displacement.

Area under the graph
The area under the graph equals the work done.
For a rectangle (constant force) the area is Fs.
For a triangle, as when stretching a spring from zero force, the area is ½Fx.
For irregular curves, count squares.
What the symbols mean:
Work done by a spring that obeys Hooke's law, F = kx, is the area of the triangle, ½kx2,
  • which is the elastic potential energy stored (link to 3.7).
    What the symbols mean: F = force (N) · k = spring constant (N m−1) · x or e = extension (m)

If the force is positive over one section and negative over another, areas above the axis count as positive work and areas below as negative.

Section 4 of 8Work and energy links

When work is done on a body, its energy changes: the work done equals the energy transferred.

1
Work done by the resultant force equals the change in kinetic energy (the work–energy theorem).
2
Work done lifting an object equals its gain in gravitational potential energy.
3
Work done against friction appears as thermal energy.
4
Power is the rate of doing work, P = W/t, and for a constant force at constant speed P = Fv.

Remember that a person holding a heavy bag stationary does no work in the physics sense, because there is no displacement, even though muscles tire from chemical energy conversions inside the body.

Section 5 of 8Practical applications of work

Work calculations are common in experimentsa trolley pulled up a slope by a falling mass, a student lifting books to a shelf, or a spring compressed by a force sensor.

1
A force sensor and a motion sensor connected to a computer can generate a force–displacement graph whose area is calculated by software.
2
For a vehicle travelling at constant speed, the engine's driving force does work against drag and friction, with power P = Fv.
3
Remember that the work done is the energy transferred, so the kinetic energy of a body changes by exactly the net work done on it.
4
This is the bridge to the energy methods in the next lesson.

In a typical practical, a force is measured with a newton-meter while a trolley is pulled a known distance along a bench; the work done is the force times the distance moved in the direction of the force.

Practical applications of work
Repeating at different angles of the string lets you test W = Fs cos θ.
Uncertainty arises mainly from the force reading fluctuating and from friction, so use a low-friction surface and take several readings.

Section 6 of 8Worked examples

  1. A 50 N force pulls a trolley 8.0 m along a floor via a rope at 35° above the horizontal. Find the work done by the force.

  2. A crane lifts a 250 kg load at constant speed through 12 m. Find the work done by the crane's cable.

  3. A force rises uniformly from 0 to 30 N as an object is moved 0.20 m. Find the work done.

Section 7 of 8Key terms

TermMeaning
Work doneEnergy transferred by a force moving through a distance in its direction.
JouleWork done by 1 N moving 1 m in the direction of the force.
Component of forcePart of the force acting along a chosen direction.
Force–displacement graphPlot whose area gives the work done.
Negative workWork done by a force opposing the displacement.
PowerRate of doing work, in watts.

Section 8 of 8Exam technique

1
Multiplying by cos θ with θ taken from the wrong direction (the angle must be between force and displacement).
2
Saying work is done on an object that does not move. No displacement, no work.
3
Using work = force × time. That is impulse. Work is force × displacement.
4
Forgetting that work can be negative.
5
Ignoring that the normal force does no work on a horizontally moving object.
Try it in the virtual labRoller coaster: energy and the loop \n