Further Dynamics

Revise Further Dynamics for Mechanics 3 WME03 (A2 Level) — revision notes and instant AI marking.

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Edexcel IAL Mechanics 3 (WME03)

Further Dynamics

The big idea: Variable forces lead to differential equations; when acceleration is proportional to displacement and towards a fixed point, the motion is simple harmonic.

Summary — What This Chapter Covers

Newton's law of gravitation: F = k/x²; at the Earth's surface mg = k/R².
SHM: ẍ = −ω²x.
v² = ω²(a² − x²), period 2π/ω, maximum speed aω, maximum acceleration aω².
A particle on an elastic string or spring performs SHM about its equilibrium position (while the string stays taut).

1. Variable Forces and Newton's Law of Gravitation

Gravitation

F = mgR²/x²

Write the force as mgR²/x², use v dv/dx for the acceleration, and integrate.

2. Simple Harmonic Motion

SHM

ẍ = −ω²x

To prove SHM, show the resultant force gives ẍ = −ω²x about a fixed centre.

Worked Example

A particle performs SHM with amplitude 0.5 m and period π s. Find its maximum speed and its speed 0.3 m from the centre.

3. Oscillations on Strings and Springs

Oscillations

SHM about equilibrium

Memorise

SHM

ẍ = −ω²x

Speed

v² = ω²(a² − x²)

Period

2π/ω

String

ω² = λ/(ml)

Common Mistakes

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Further Dynamics