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Mechanics · Work, energy & power

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Work done by a force

The work done by a constant force, W = Fd when the force is along the motion and W = Fd cosθ when it acts at an angle, together with work done against gravity and against resistance — and the sign convention that makes a force opposing the motion do negative work.

Mechanics · Work, energy & power

Work done by a force

The work done by a constant force, W = Fd when the force is along the motion and W = Fd cosθ when it acts at an angle, together with work done against gravity and against resistance — and the sign convention that makes a force opposing the motion do negative work.

Why it works

When a constant force moves its point of application through a displacement, it does work. If the force FF acts along the direction of motion and the object moves a distance dd, the work done is W=Fd,W = Fd, measured in joules (1 J=1 N m1\text{ J} = 1\text{ N m}). Work is a scalar — it has no direction — but it does carry a sign.

Force at an angle. If the force acts at an angle θ\theta to the direction of motion, only the component along the motion does work. That component is Fcos⁡θF\cos\theta, so W=Fdcos⁡θ.W = Fd\cos\theta. (You never need the scalar product for this in Paper 4 — resolving the force with cos⁡θ\cos\theta is all that is required.) When θ=0\theta = 0 the force is along the motion and W=FdW = Fd; when θ=90∘\theta = 90^\circ the force is perpendicular to the motion and does no work — which is why the normal contact force and the weight of an object moving horizontally contribute nothing.

Work done against gravity. Raising a mass mm through a vertical height hh requires work against its weight mgmg: W=mgh.W = mgh. Only the vertical height matters, not the path taken. In this course take g=10 m s−2g = 10\text{ m s}^{-2}.

The sign of work. A force whose component points the same way as the motion does positive work (it adds energy); a force whose component opposes the motion does negative work (it removes energy). Friction and other resistances always oppose motion, so the work done by a resistance is negative — equivalently, the work done against the resistance is positive and equals (resistance) ×\times (distance).

Net work. When several forces act, the total (net) work is the sum of the works done by each force — which is the same as the work done by the resultant force. On a rough floor, Wnet=(driving force−friction)×dW_{\text{net}} = (\text{driving force} - \text{friction}) \times d.