A-level Mathematics/OCR/M3

This is the Mechanics 3 module of the A-level Mathematics book. This module assumes that you are familiar with the C1, C2, C3, C4, M1 and M2 modules. Please post all questions to A-level Mathematics/Discussion.

Formulae

By the end of this module you will be expected to have learnt the following formulae:

Kinematics

• ${\displaystyle a=v{\frac {dv}{dx}}}$

Simple Harmonic Motion

• ${\displaystyle {\ddot {x}}=-\omega ^{2}x}$
• ${\displaystyle v^{2}=\omega ^{2}(a^{2}-x^{2})\,}$
• ${\displaystyle T={\frac {2\pi }{\omega }}}$
• ${\displaystyle x=a\sin(\omega t)\,}$ or ${\displaystyle x=a\cos(\omega t)\,}$ or, generally, ${\displaystyle x=a\cos(\omega t+\epsilon )\,}$

Elastic Springs and Strings

• Hooke's Law: ${\displaystyle T={\frac {\lambda x}{l}}}$
• Elastic potential energy: ${\displaystyle E_{p}={\frac {\lambda x^{2}}{2l}}}$

Small-angle Approximations

• ${\displaystyle \sin \theta \approx \theta }$
• ${\displaystyle \cos \theta \approx 1-{\frac {1}{2}}\theta ^{2}}$
• ${\displaystyle \tan \theta \approx \theta }$

only applies when ${\displaystyle \theta }$ is in radians