Problems

An approximate value for the angle \(\phi\), measured in radians, is \(\displaystyle\frac{3 \sin\phi}{2 + \cos\phi}\), provided \(\phi\) is less than \(\frac{1}{2}\pi\). Establish this result when \(\phi\) is small, and shew that the error is approximately \(\displaystyle\frac{\phi^5}{180}\). Hence, express approximately the acute angles of a right-angled triangle in terms of the sides, and deduce that \[ \frac{6(a+b)c + 3c^2}{ab + 2(a+b)c + 4c^2} \] is nearly equal to \(\frac{1}{2}\pi\) for all positive values of \(a\) and \(b\), provided \(c^2 = a^2+b^2\).

\(n\) quantities are given. \(s_r\) denotes the sum of the products of all combinations of the quantities \(r\) at a time. Prove that the sum of the products of all combinations of the squares of the quantities \(m\) at a time is given by \[ s_m^2 - 2s_{m-1}s_{m+1} + 2s_{m-2}s_{m+2} + \dots + (-)^{m-1}2s_1s_{2m-1} + (-)^m 2s_{2m} \] where we suppose \(2m \le n\).

Shew that, by plotting a curve connecting the reciprocal of the acceleration of a body with its velocity, it is possible to estimate the time required for a given change of velocity. The acceleration of a tramcar starting from rest decreases by an amount proportional to the increase of speed, from 1.5 f.s.s. at starting to 0.5 f.s.s. when the speed is 5 m.p.h. Find the time taken to reach 5 m.p.h. from rest.

Shew that a system of particles has one and only one centre of mass. Find the centres of mass of the following solids of uniform density, (1) a tetrahedron; (2) a pyramid with a plane base. A solid is bounded by five faces, namely, a parallelogram \(ABCD\), a trapezium \(ABFE\) in which \(AB, EF\) are parallel, a trapezium \(CDEF\), a triangle \(AED\) and a triangle \(BFC\). Prove that the centre of mass of the solid divides the join of the middle point of \(EF\) and the centre of the parallelogram \(ABCD\) in the ratio \(3AB+EF : AB+EF\).

A cyclic quadrilateral \(ABCD\) is such that a circle can also be inscribed in it. If the sides \(AB, BC, CD, DA\) are equal to \(a, b, c, d,\) respectively, prove that \[ \cos A = \frac{ad-bc}{ad+bc}. \]

Prove the formulae

1. [(i)] \(\Delta = \frac{s^2}{\cot\frac{1}{2}A + \cot\frac{1}{2}B + \cot\frac{1}{2}C}\),
2. [(ii)] \(\Delta = s^2 \tan\frac{1}{2}A \tan\frac{1}{2}B \tan\frac{1}{2}C\),

for the area of a triangle \(ABC\), \(s\) being the semi-perimeter. Deduce that \(\Delta/\Delta_0 = p^2\), where \(\Delta_0\) is the area of the equilateral triangle with the same perimeter as \(ABC\), and \(p\) the ratio of the geometric to the arithmetic means of \(\cot\frac{1}{2}A, \cot\frac{1}{2}B, \cot\frac{1}{2}C\). Hence, or otherwise, prove that a triangle whose perimeter is given has its maximum area when it is equilateral.

A particle is projected vertically upwards with velocity \(V\), and the resistance of the air produces a retardation \(kv^2\), where \(v\) is the velocity. Shew that the velocity \(V'\) with which the particle will return to the point of projection is given by \[ \frac{1}{V'^2} = \frac{1}{V^2} + \frac{k}{g}. \]

Explain what is meant by the shearing stress and bending moment in a beam, and obtain the relations between them and the load. Discuss the characteristics of the distributions of shearing stress and bending moment along a beam which is loaded at various points and supported in a horizontal position at two points when the beam is (a) light, (b) uniformly heavy. In case (a) shew that the funicular polygon for the loads and reactions is also a bending moment diagram such that an intercept by the polygon on a line parallel to the loads is a measure of the bending moment at the corresponding point of the beam.

\(AB, BC\) are adjacent sides of a regular polygon, \(O, D\) are the middle points of \(AB, BC\), respectively. By taking coordinate axes at \(O\), so that the axis of \(x\) is along \(OB\), prove that the line joining the centre of the polygon to the centroid of the triangle \(ABD\) is perpendicular to \(AD\).

Prove the identities

1. [(i)] \(\displaystyle\sum_{v=1}^n (2v-1)\sin(2v-1)\theta = \frac{\cos\theta\sin 2n\theta - 2n\sin\theta\cos 2n\theta}{2\sin^2\theta}\).
2. [(ii)] \(\displaystyle\sum_{\alpha,\beta,\gamma} \sin^2\alpha\sin(\beta-\gamma)\cos(\theta-\beta)\cos(\theta-\gamma) = -\sin(\beta-\gamma)\sin(\gamma-\alpha)\sin(\alpha-\beta)\cos^2\theta\).