Work

What is work?

A force does work when its point of application moves with a component along or against the force. For a constant force, multiply its component along the total displacement by the displacement’s magnitude; for a varying force, add the contributions along the actual path. Work is measured in a specified reference frame.

For a constant force $\vec F$ and a displacement $\vec d$ at an angle θ to each other:

Here F cos θ is the component of the force along the displacement. Work is the scalar product of force and displacement, so it is a scalar: it has a size and a sign, but no direction.

Positive, negative and zero work

The sign of the work comes from cos θ:

Positive work is done by a force that helps the motion: it feeds energy into the body. Negative work is done by a force that opposes the motion, such as friction, which usually acts at 180° to the displacement, so cos 180° = −1.

No work is done in three cases:

1. The point of application stays at rest. Pushing hard against a brick wall, or a weightlifter holding a heavy weight still above the head, does no work on the wall or the weight, however tiring it is. The tiredness comes from muscles working inside the body, not from work done on the load.
2. Zero force. A block gliding on a smooth table has no horizontal force on it, yet it may travel a long way.
3. Force perpendicular to the displacement. Gravity does no work on a block moving along a level table. If the Moon’s orbit is taken as a circle, the Earth’s pull, along the radius, is always at right angles to the Moon’s motion, along the tangent, so it does no work. The same is true of any centripetal force in uniform circular motion.

A force does no work on a point that remains at rest. Returning to the starting point is different: kinetic friction can do negative work throughout a closed path even though the total displacement is zero. A constant force, by contrast, does zero total work over any closed path.

Two cautions about signs. The normal force can do work: when a lift moves up, the floor’s push on you is along your displacement and does positive work. And friction is not always negative: static friction on a crate carried by an accelerating truck acts forward, along the crate’s motion, and does positive work on it.

Unit and dimensions of work

From W = Fd cos θ, the dimensions of work are those of force times length:

Work and energy have the same dimensions. Their SI unit is the joule (J), named after the British physicist James Prescott Joule:

One joule is the work done when a force of 1 N moves its point of application through 1 m along the force. Some other units of work and energy:

The calorie conversion uses the thermochemical convention in NIST Special Publication 811, Appendix B.9; other calorie conventions have slightly different values.

Work done by more than one force

When several forces act on a body, each does its own work, with its own sign. The net work is the sum of these, and, for a body treated as a particle, it equals the work done by the net force. By the work-energy theorem, the net work is the change in the body’s kinetic energy.

A lesson from the third law is worth noting. When a cyclist skids to a stop, the road pushes back on the cycle and does negative work on it. The cycle pushes on the road with an equal and opposite force, but the road does not move, so that force does no work. NCERT Example 5.3 (page 74) works this out: a 200 N stopping force over 10 m does −2000 J of work on the cycle, while the cycle does no work on the road. Equal and opposite forces do not always do equal and opposite work.

Work in daily life

• Lifting a load. Lifting a 10 kg bag through 1 m at a steady speed needs an upward force equal to its weight, about 98 N, so you do about 98 J of work on it, while gravity does about −98 J.
• Pulling with a rope at an angle. Only the part of your pull along the ground does work on a sledge. The upward part does no work, because the sledge does not move upward; it only makes the sledge press less on the ground, which reduces friction.
• Carrying a bag along a level corridor. At a steady speed, the upward force of your hand on the bag is at right angles to its motion, so it does no work on the bag.
• Brakes and friction. They do negative work, taking kinetic energy away and turning it into heat.

Solved examples

Common mistakes to avoid

Quick revision summary

• Work done by a constant force is the component of the force along the displacement times the size of the displacement.
• Work is a scalar. It is positive when the angle is less than 90°, zero at 90° and negative when the angle is more than 90°.
• A force does no work if its point of application stays at rest, if the force is zero, or if it is perpendicular to the motion at every instant. Zero total displacement alone is not enough for a varying force.
• The SI unit of work and energy is the joule, 1 J = 1 N m; the dimensions are [M L^² T^⁻²].
• The net work is the sum of the works of all the forces; equal and opposite forces need not do equal and opposite work.

Questions students ask

Is work a scalar or a vector quantity?

Work is a scalar quantity. For a constant force it is the scalar (dot) product of two vectors, W = F · d = Fd cos θ, so it has a size and a sign (positive, negative or zero) but no direction.

What is the difference between work and energy?

Energy is the capacity of a body to do work, something the body has; work is energy transferred to or from a body by a force acting through a displacement, something that happens. They have the same SI unit, the joule, and the same dimensions, [M L^² T^⁻²]. By the work-energy theorem, the net work done on a body equals the change in its kinetic energy.

What is the work done by gravity when a body is lifted or lowered?

When a body of mass m is raised through a height h near the Earth’s surface, gravity does −mgh of work; when it comes down through h, gravity does +mgh. The work depends only on the change in height, not on the path, because gravity is a conservative force (see path independence). If you lift the body at a steady speed, you do +mgh of work against gravity.

What is the work done by friction?

The work done by kinetic friction on a body sliding a distance d over a fixed surface is W = −μ_kNd, because the friction force μ_kN acts against the motion; on a level floor with only horizontal pushes, N = mg and W = −μ_kmgd. This negative work takes kinetic energy away and turns it into heat. Friction’s work is not always negative, though: a box dropped onto a moving conveyor belt is dragged forward by friction, which does positive work on it until the box moves with the belt.

Why is the work done by a centripetal force zero?

The work done by a centripetal force is zero because the force points along the radius, at right angles to the velocity, which is along the tangent. With θ = 90°, Fd cos 90° = 0 for every small step, so a body in uniform circular motion keeps its kinetic energy. For the same reason, the magnetic force on a moving charge, always perpendicular to its velocity, does no work.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. A force of 20 N acts on a body that moves 4 m in a direction making 60° with the force. Find the work done.

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   $$\begin{aligned} W &= 20 \times 4 \times \cos 60^\circ \\ &= 40\ \text{J} \end{aligned}$$

2. Q2. A stone tied to a string moves in a circle at constant speed on a smooth horizontal table. How much work does the tension do in one full revolution? Explain.

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   Zero. The tension acts along the string, towards the centre, while the stone moves along the tangent. The force is always at right angles to the displacement, so it does no work at any moment.

3. Q3. A 5 kg block is dragged 8 m at constant speed along a rough level floor by a horizontal force. The coefficient of kinetic friction is 0.25. Find the work done by the applied force and by friction. Take g = 9.8 m s⁻².

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   At constant speed the applied force equals friction:

   $$\begin{aligned} f &= 0.25 \times 5 \times 9.8 \\ &= 12.25\ \text{N} \end{aligned}$$

   The applied force does 12.25 × 8, that is 98 J, and friction does −98 J. The net work is zero, so the speed does not change.