Summary — What This Chapter Covers
- Work done by a constant force F moving its point of application a distance d at angle θ to the force: W = Fd cos θ.
- Kinetic energy ½mv2; gravitational potential energy mgh (h vertical).
- Work–energy principle: work done by driving force − work done against resistance = gain in KE + gain in PE.
- If only gravity does work, mechanical energy is conserved — even on a curved path.
- Power = rate of doing work; for a force in the direction of motion, P = Fv.
1. Work Done and Energy
Work
W = Fd cos θ
The work done by a constant force is the force multiplied by the distance moved in the direction of the force: if the force is at angle θ to the displacement, W = Fd cos θ. Work done against gravity in raising a mass m through a vertical height h is mgh; work done against friction F over a distance d is Fd. Work and energy are measured in joules (J).
Kinetic energy is ½mv2. Gravitational potential energy is mgh relative to a chosen level — only changes in PE matter, so choose any convenient zero level.
Work and energy
W = Fd cos θ KE = ½mv2 PE = mgh
Worked Example
A sledge is pulled 12 m across horizontal snow by a rope at 40° above the horizontal with tension 30 N. Find the work done by the tension.
Practice Question 1
A crate of mass 50 kg is pushed 8 m up a slope inclined at angle α where sin α = 0.6. Find the gain in potential energy.
2. Work–Energy Principle
Energy
Balance the energy books
For the whole motion: initial KE + initial PE + work done by driving forces = final KE + final PE + work done against resistances. The normal reaction does no work (it is perpendicular to the motion). If there are no resistances and no driving forces — a smooth surface, even a curved slide — mechanical energy is conserved, and only the vertical drop matters.
Energy methods are the natural choice when the question asks about speed and distance but not time, or when the path is curved and the forces are not constant.
Work–energy principle
KE1 + PE1 + Wdriving = KE2 + PE2 + Wresistance
Worked Example 1
A child starts down a smooth curved slide whose top is 5 m above the bottom with speed 2 m s−1. Find the speed at the bottom.
Worked Example 2
A block of mass 3 kg is released from rest and slides 6 m down a rough plane inclined at 30°. The friction force is 9 N. Find its speed after 6 m.
3. Power
Power
P = Fv
Power is the rate at which work is done, measured in watts (1 W = 1 J s−1). Average power = work done ÷ time taken. For a vehicle whose engine exerts a driving force F at speed v, P = Fv, so the driving force at a given speed is F = P/v.
P/v − resistance − mg sin α = ma
maximum (steady) speed
Power