25 Mechanics 2

A knowledge of the topics and associated formulae from Modules Core 1 – Core 4, and Mechanics 1 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.

 Centres of Mass $\displaystyle\bar{X} \Sigma m_i = \Sigma m_i x_i \space$ and $\displaystyle \space \bar{Y} \Sigma m_i = \Sigma m_i y_i$ Circular Motion $\displaystyle v = r \omega, \space a = r \omega^2 \space$ and $\displaystyle \space a = \frac{v^2}{r}$ Work and Energy Work done, constant force:   $\text{Work}= Fd \cos \theta$ Work done, variable force in direction of motion in a straight line: $\text{Work}= \int \mbox{F}\space \mbox{d}x$ Gravitational Potential Energy $= mgh$ Kinetic Energy $= \frac{1}{2}mv^2$ Elastic potential energy $\displaystyle = \frac{\lambda}{2l}e^2$ Hooke’s Law $\displaystyle T = \frac{\lambda}{l}e$
 The application of mathematical modelling to situations that relate to the topics covered in this module.
 Finding the moment of a force about a given point. Knowledge that when a rigid body is in equilibrium, the resultant force and the resultant moment are both zero. Determining the forces acting on a rigid body when in equilibrium. This will include situations where all the forces are parallel, as on a horizontal beam or where the forces act in two dimensions, as on a ladder leaning against a wall. Centres of Mass. Integration methods are not required. Finding centres of mass by symmetry (eg for circle, rectangle). Finding the centre of mass of a system of particles. Centre of mass of a system of particles is given by   $(\bar{X}, \bar{Y})$   where  $\bar{X} \Sigma m_i = \Sigma m_i x_i \space$ and $\space \bar{Y} \Sigma m_i = \Sigma m_i y_i$ Finding the centre of mass of a composite body. Finding the position of a body when suspended from a given point and in equilibrium.
 Relationship between position, velocity and acceleration in one, two or three dimensions, involving variable acceleration. Application of calculus techniques will be required to solve problems. Finding position, velocity andacceleration vectors, by thedifferentiation or integrationof   $\mbox{f}(t){\bf \mbox{i}} + \mbox{g}(t){\bf \mbox{j}} + \mbox{h}(t){\bf\mbox{k}}$, with respect to $\space t$. If       $\space \space \space \space \space{\bf \mbox{r}} = \mbox{f}(t){\bf \mbox{i}} + \mbox{g}(t){\bf \mbox{j}} + \mbox{h}(t){\bf \mbox{k}}$ then       ${\bf \mbox{v}} = \mbox{f}'(t){\bf \mbox{i}} + \mbox{g}'(t){\bf \mbox{j}} + \mbox{h}'(t){\bf \mbox{k}}$ then       ${\bf \mbox{a}} = \mbox{f}''(t){\bf \mbox{i}} + \mbox{g}''(t){\bf \mbox{j}} + \mbox{h}''(t){\bf \mbox{k}}$ Vectors may be expressed in the form $\space \mbox{a}{\bf\mbox{i}}+\mbox{b}{\bf \mbox{j}}+\mbox{c}{\bf \mbox{k}}$ as columnvectors. Candidates may use either notation.
 Application of Newton’s laws to situations, with variable acceleration. Problems will be posed in one, two or three dimensions and may require the use of integration or differentiation.
 One-dimensional problemswhere simple differential equations are formed as a result of the application of Newton’s second law. Use of  $\displaystyle \frac{\mbox{d}v}{\mbox{d}t}$  for acceleration, to form simple differential equations, for example, $\displaystyle \space m\frac{\mbox{d}v}{\mbox{d}t} = -\frac{k}{\sqrt{v}} \space \text{or} \space m\frac{\mbox{d}v}{\mbox{d}t} = k(v-2)$. Use of $\space \displaystyle a=\frac{\mbox{d}^2 x}{\mbox{d}t^2} = \frac{\mbox{d}v}{\mbox{d}t};\space v = \frac{\mbox{d}x}{\mbox{d}t}$. The use of  $\displaystyle \frac{\mbox{d}v}{\mbox{d}t} = v\frac{\mbox{d}v}{\mbox{d}x}$ is not required. Problems will require the use of the method of separation of variables.
 Motion of a particle in a circle with constant speed. Problems will involve either horizontal circles or situations, such as a satellite describing a circular orbit, where the gravitational force is towards the centre of the circle. Knowledge and use of the relationships $\displaystyle {v} = r \omega, \space {a} = r \omega^2 \space$ and $\displaystyle \space a=\frac{v^2}{r}$. Angular speed in radians $\space \mbox{s}^{-1}$   converted from other unitssuch as revolutions per minute or time for one revolution. Use of the term angular speed. Position, velocity and acceleration vectors in relation to circular motion in terms of $\space{\bf \mbox{i}} \space$ and $\space {\bf \mbox{j}}$. Candidates may be required to show that motion is circular by showing that the body is at a constant distance from a given point. Conical pendulum.

 Work done by a constant force. Forces may or may not act in the direction of motion. Work done$\space=Fd \cos \theta$ Gravitational potential energy. Universal law of gravitation will not be required. Gravitational Potential Energy $= mgh$ Kinetic energy. Kinetic Energy $=\frac{1}{2}mv^2$ The work-energy principle. Use of Work Done = Change in Kinetic Energy. Conservation of mechanical energy. Solution of problems using conservation of energy. One-dimensional problems only for variable forces. Work done by a variable force. Use of $\int F \space \mbox {d}x$ will only be used for elastic strings and springs. Hooke’s law. $\displaystyle T = \frac{\lambda}{l}e$. Elastic potential energy for strings and springs. Candidates will be expected to quote the formula for elastic potential energy unless explicitly asked to derive it. Power, as the rate at which a force does work, and the relationship $\space P = Fv$.
 Circular motion in a vertical plane. Includes conditions to complete vertical circles.