27 Mechanics 4

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.

 Rotations Moment of a Force $=\bf \mbox{r} \times \mbox{F}$ Moment of Inertia $\displaystyle = \sum_{i=1}^n m_ix_i^2$ Rational Kinetic Energy $= \frac{1}{2}I \omega^2 \space \text{or} \space \frac{1}{2}I \dot{\theta}^2$ Resultant Moment $= I \ddot{\theta}$ Centre of Mass For a Uniform Lamina $\displaystyle \bar{x} =\frac {\int_a^b xy \space\mbox{d}x} {\int_a^b y \space\mbox{d}x} \space \space \space$ $\displaystyle \bar{y} = \frac{\int_a^b \frac{1}{2} y^2 \space\mbox{d}x}{\int_a^b y \space\mbox{d}x}$ For a Solid of Revolution (about the $x$-axis) $\displaystyle \bar{x} = \frac{\int_a^b \pi xy^2 \space\mbox{d}x}{\int_a^b \pi y^2 \space\mbox{d}x}$
 Couples. Understanding of the concept of a couple. Reduction of systems of coplanar forces. Reduction to a single force, a single couple or to a couple and a force acting at a point. The line of action of a resultant force may be required. Conditions for sliding and toppling. Determining how equilibrium will be broken in situations, such as a force applied to a solid on a horizontal surface or on an inclined plane with an increasing slope. Derivation of inequalities that must be satisfied for equilibrium.
 Finding unknown forces acting on a framework. Finding the forces in the members of a light, smoothly jointed framework. Determining whether rods are in tension or compression. Awareness of assumptions made when solving framework problems.
 The vector product $\bf\mbox{i} \times \mbox{i} = \mbox{0}\mathrm{,}\space \mbox{i} \times \mbox{j}=\mbox{k}\mathrm{,}\space \mbox{j} \times \mbox{i} = -\mbox{k}$ etc Candidates may use determinants to find vector products. The result $|\bf\mbox{a} \times \mbox{b}| = |\mbox{a}|| \mbox{b}|\sin \theta$ The moment of a force as $\bf \mbox{r} \times \mbox{F}$. Vector methods for resultant force and moment. Application to simple problems. Finding condition for equilibrium, unknown forces or points of application.
 Centre of mass of a uniform lamina by integration. Centre of mass of a uniform solid formed by rotating a region about the $x$-axis. Finding $x$ and $y$ coordinates of the centre of mass.
 Moments of inertia for a system of particles. About any axis $\displaystyle I = \sum_{i=1}^n m_i x_i^2$ Moments of inertia for uniform bodies by integration. Candidates should be able to derive standard results, ie rod, rectangular lamina, hollow or solid sphere and cylinder. Moments of inertia of composite bodies. Bodies formed from simple shapes. Parallel and perpendicular axis theorems. Application to finding moments of inertia about different axes.
 Angular velocity and acceleration of a rigid body. To exclude small oscillations of a compound pendulum. Motion of a rigid body about a fixed horizontal or vertical axis. $I \ddot{\theta} =$ Resultant Moment Including motion under the action of a couple. Rotational kinetic energy and the principle of conservation of energy. Rotational Kinetic Energy $= \frac{1}{2} I \omega^2 \space$ and $\space \frac{1}{2} I \dot{\theta}^2$ To include problems such as the motion of a mass falling under gravity while fixed to the end of a light inextensible string wound round a pulley of given moment of inertia. Moment of momentum (angular momentum). The principle of conservation of angular momentum. To include simple collision problems, eg a particle colliding with a rod rotating about a fixed axis. Forces acting on the axis of rotation