Problems

Shew that

1. [(i)] \(1+2(\cos\alpha+\cos 2\alpha+\dots+\cos n\alpha) = \sin(n+\frac{1}{2})\alpha \operatorname{cosec} \frac{\alpha}{2}\).
2. [(ii)] \(1+2(\cos\alpha\cos\theta+\cos 2\alpha\cos 2\theta+\dots+\cos n\alpha\cos n\theta) = \frac{\cos n\alpha \cos(n+1)\theta - \cos(n+1)\alpha\cos n\theta}{\cos\theta-\cos\alpha}\).

Deduce that \begin{align*} 1^2\cos\theta+2^2\cos 2\theta+\dots+n^2\cos n\theta \\ = \tfrac{1}{2}\operatorname{cosec}^2\tfrac{1}{2}\theta\{(n+\tfrac{1}{2})^2\cos n\theta - n^2\cos(n+1)\theta - \sin(n+\tfrac{1}{2})\theta\operatorname{cosec}\tfrac{1}{2}\theta\}. \end{align*}

Prove that the following definitions of the curvature of a curve at a point \(P\) lead to the same value.

1. [(i)] The rate, in radians per unit of arc, at which the tangent turns.
2. [(ii)] The limit to which the reciprocal of the radius of a circle touching the curve at \(P\) and passing through an adjacent point \(Q\) tends, as \(Q\) tends to \(P\).

Examine the nature of the evolute of a given curve in the neighbourhood of the following points on the given curve

1. [(a)] a point of inflexion,
2. [(b)] a cusp,
3. [(c)] a point of maximum or minimum curvature.

Trace the curve \(b(ay-x^2)^2=x^5\), and shew that the evolute has a point of inflexion corresponding to the origin on the curve.

Give some account of the theory of a framework of rods, dealing with (i) the number of rods necessary for a frame with \(n\) joints to be just rigid, (ii) the graphical determination of stresses (\(\alpha\)) in a `simple' frame under given forces at the joints, (\(\beta\)) in a non-simple frame. \par Illustrate by the two cases of a pentagon \(ABCDE\) of rods jointed at the corners, (a) with connecting rods \(AC, CE\), (b) with connecting rods \(AD, CE\), in equilibrium under three given forces at the corners \(B, C, E\).

A thin wire has the form of a circle in a vertical plane with centre \(C\). \(A, B\) are pegs attached to the wire so that \(CA, CB\) make angles \(\alpha\) on opposite sides of the downward vertical through \(C\). A small ring of mass \(M\) can slide on the wire, and is attached to two strings passed over the pegs with masses \(m\) hanging from their ends. Write down the potential energy of the system when the radius to \(M\) makes an angle \(\theta\) with the vertical. \par Hence discuss the stability of equilibrium positions in the cases \(M \gtrless m\sin\frac{1}{2}\alpha\).

A particle of mass \(m\) is attached by a string to a point on the circumference of a fixed circular cylinder of radius \(a\) whose axis is vertical, the string being initially horizontal and tangential to the cylinder. The particle is projected with velocity \(v\) at right angles to the string along a smooth horizontal plane so that the string winds itself round the cylinder. \par Shew (i) that the velocity of the particle is constant, \par (ii) that the tension in the string is inversely proportional to the length which remains straight at any moment, \par (iii) that if the initial length of the string is \(l\) and the greatest tension the string can bear is \(T\), the string will break when it has turned through an angle \(l/a-mv^2/aT\).

Shew that if a number of particles connected by inelastic strings move under no forces, their linear momentum and energy are constant. \par Three equal particles \(A, B, C\) connected by inelastic strings \(AB, BC\) of length \(a\) lie at rest with the strings in a straight line on a smooth horizontal table. \(B\) is projected with velocity \(V\) at right angles to \(AB\). Shew that the particles \(A\) and \(C\) afterwards collide with relative velocity \(\frac{2V}{\sqrt{3}}\). \par If the coefficient of restitution is \(e\), find the velocities of the three particles when the string is again straight.

Shew how to find graphically the resultant of any number of given coplanar forces. \par A uniform plank of weight 30 lbs. and length 12 feet is supported at each end and carries two weights of 10 lbs. each at distances 4 and 5 feet from one end. Find graphically the pressures on the supports.

Two particles of a system of masses \(m_1, m_2\) are at \(A, B\). If these two particles are interchanged, prove that the centre of gravity of the whole system moves through a distance \(\frac{m_1-m_2}{\Sigma m} AB\) parallel to \(AB\).

Two light rods are freely jointed together at one end and the other ends carry weights \(W, W'\). The rods are in a vertical plane, each being supported by one of two smooth pegs on the same level. If there is equilibrium when the rods are at right angles, prove that \[ a^2W^2+b^2W'^2=c^2(W+W')^2, \] where \(a, b\) are the lengths of the rods and \(c\) the distance between the pegs.

Two cylinders of equal radius but different weights \(W, W' (W'>W)\) rest inside another cylinder which is fixed. All the cylinders are equally rough, their axes are horizontal and the line of contact of the two smaller cylinders is vertically below the axis of the fixed cylinder. Shew that equilibrium is impossible unless the coefficient of friction is \(> \frac{W'-W}{W'+W}\frac{\cos\alpha}{1+\sin\alpha}\) when \(2\alpha\) is the angle between the planes through the axis of the fixed cylinder and the axes of the smaller cylinders. Shew also that if the coefficient of friction has this value each of the two smaller cylinders will be on the point of rolling on the fixed cylinder.