Further Mechanics 2

What further mechanics 2 covers

One of the eight optional papers of 9FM0, sat as Paper 4C. It is an Option 2 paper, so it may be taken only in a matching pair with Further Mechanics 1. It covers circular motion, where a body balances and when it topples, and the oscillations that follow when a force pulls back in proportion to displacement.

The main ideas

• Angular speed, its link to linear speed and period, and radial acceleration in both of its forms.
• Horizontal circular motion, with the conical pendulum and the banked track as the standard settings.
• Vertical circles, combining conservation of energy for the speed with the radial equation for the tension, and the conditions for completing one.
• Centres of mass of discrete distributions, of composite laminae with pieces removed as negative areas, and of frameworks of rods.
• Centres of mass by integration, for laminae, solids of revolution and bodies of varying density.
• Equilibrium of rigid bodies: a suspended lamina, and whether a body on a rough slope topples or slides first.
• Newton's laws with a variable force, simple harmonic motion, oscillations on strings and springs, and acceleration given as a function of x, t or v.

The results it turns on

v = rω, with radial acceleration rω² or v²/r

uniform motion in a circle

tan θ = v²/rg

the design speed of a banked track, where the reaction alone suffices

from the lowest point, u² ≥ 5gr completes the circle and u² ≤ 2gr oscillates

the two boundary cases on a string

x_G = Σ(mx)/Σm

the centre of mass as a mass-weighted average

x_G = ∫xy dx ÷ ∫y dx, and y_G = ∫½y² dx ÷ ∫y dx

a lamina by integration, where the two coordinates differ

acceleration = −ω²x, with v² = ω²(a² − x²) and period 2π/ω

simple harmonic motion

Where it usually goes wrong

• A string or the inside of a track can act inwards only, so it needs the square of the speed to be at least gr at the top. A rod or a wire can also push, and needs only a positive speed there.
• For a framework, each rod is weighted by its length at its own midpoint. A wire and a lamina of the same outline therefore have different centres of mass in general.
• Toppling uses half the base width over the height of the centre of mass, and the smaller of the toppling and sliding angles is the one that happens, so both are computed and compared.
• Choose v dv/dx when the force depends on position and dv/dt when it depends on time. The suvat equations are unavailable either way, and saying so earns the mark.

Where to start

Circular motion first, horizontal then vertical. The three centre of mass lessons form one block, discrete then plane figures then integration, with equilibrium and toppling directly after them. Variable force, simple harmonic motion and further kinematics close the paper, and the oscillation work assumes Hooke's law from Further Mechanics 1.