# 28 Mechanics 5

A knowledge of the topics and associated formulae from Modules Core 1 – Core 4, Mechanics 1 and Mechanics 2 is required. Candidates may use relevant formulae included in the formulae booklet without proof.
Candidates should learn the following formulae, which are not included in the formulae booklet, but which may be required to answer questions.

 Energy Formulae $\mbox{PE} = mgh \space\space\space$Gravitational Potential Energy $\mbox{EPE} = \frac{1}{2} k \space e^2 \space\space\space$ or $\space \displaystyle \frac{\lambda}{2l}e^2\space \space \space$ Elastic Potential Energy Simple Harmonic Motion $\displaystyle v^2 = \omega^2(a^2-x^2)$ $\displaystyle \frac{\mbox{d}^2x}{\mbox{d}t^2} = - \omega^2 x$
 Knowledge of the definition of simple harmonic motion. Finding frequency, period and amplitude Knowledge and use of the formula $\space v^2 = \omega^2 (a^2-x^2)$. Formation of simple second order differential equations to show that simple harmonic motion takes place. Problems will be set involving elastic strings and springs. Candidates will be required to be familiar with both modulus of elasticity and stiffness. They should be aware of and understand the relationship $\displaystyle \space k = \frac{\lambda}{l}$. Solution of second order differential equations of the form $\displaystyle \frac{\mbox{d}^2x}{\mbox{d}t^2}= -\omega^2x$ State solutions in the form $\displaystyle \space x = A\cos(\omega t + \alpha)$ or $\displaystyle \space x = A \cos(\omega t) + B \sin( \omega t) \space$and use these in problems. Simple Pendulum. Formation and solution of the differential equation, including the use of a small angle approximation. Finding the period.
 Understanding the terms forcing and damping and solution of problems involving them. Candidates should be able to set up and solve differential equations in situations involving damping and forcing. Damping will be proportional only to velocity. Forcing forces will be simple polynomials or of the form $\displaystyle a \sin(\omega t + \alpha),\space \omega a \sin t + b \cos \omega t \space$ or $\displaystyle \space ae^{bt}$. Light, critical and heavy damping. Candidates should be able to determine which of these will take place. Resonance. Solutions may be required for the case where the forcing frequency is equal to the natural frequency. Application to spring/mass systems.
 Finding and determining whether positions of equilibrium are stable or unstable. Use of potential energy methods. Problems will involve gravitational and elastic potential energy.
 Equation of motion for variable mass. Derive equations of motion for variable mass problems, for example, a rocket burning fuel, or a falling raindrop. Rocket problems will be set in zero or constant gravitational fields.
 Polar coordinates Transverse and radial components of velocity in polar form. These results may be stated. No proof will be required. Transverse and radial components of acceleration in polar form. Application of polar form of velocity and acceleration. Application to simple central forces. No specific knowledge of planetary motion will be required.