Further Mechanics 1

What further mechanics 1 covers

One of the eight optional papers of 9FM0, sat as Paper 3C, so revise it only if it is one of the two options you take. It covers momentum and impulse, the energy stored in a stretched string, and what happens when two spheres collide. It is also the gateway to Further Mechanics 2, which may be taken only as a matching pair with it, and it assumes the mechanics from Paper 3 of A level Mathematics.

The main ideas

• Momentum and impulse, the impulse-momentum principle, and an average force recovered from a contact time.
• Conservation of momentum in a direct collision, with coalescence and explosion as special cases of one principle.
• Impulse and momentum in vector form, with the components handled separately and the magnitude taken last.
• Work done by a force at an angle, the work-energy principle, and conservation of mechanical energy where no resistance acts.
• Power, with the driving force falling as speed rises and maximum speed reached where it has dropped to the resistance.
• Hooke's law for elastic strings and springs, and the energy stored in a stretched string or a compressed spring.
• Newton's law of restitution: direct impact, impact with a wall, successive impacts, and oblique impact resolved along the line of centres.

The results it turns on

impulse = Ft = mv − mu

the impulse-momentum principle, in newton seconds

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

conservation of momentum in a direct collision

work = Fd cos θ, and P = Fv

work done by a force at an angle, and power at a given speed

T = λx/l

Hooke's law, with x the extension and lambda a modulus in newtons

elastic energy = λx²/(2l)

the triangle under the Hooke's law line

separation speed = e × approach speed, with 0 ≤ e ≤ 1

Newton's law of restitution

Where it usually goes wrong

• Impulse is a vector, so a rebound needs an impulse matching the sum of the two speeds rather than their difference. Choose a positive direction and mark it on the diagram first.
• The normal reaction does no work, so it stays out of every energy equation however prominent it is on the force diagram.
• Against a fixed wall only restitution applies. Momentum is not conserved for the ball considered on its own, because the wall takes the rest.
• In an oblique impact on a smooth surface the coefficient multiplies the perpendicular component only, and the parallel component passes through unchanged.

Where to start

Momentum and impulse first, in one dimension and then as vectors. Work, energy and power next, then Hooke's law and elastic energy as a pair. Restitution last, with direct impact before successive impacts and oblique impact, since each of those adds one idea to the one before.