A train of mass 300 tons has a driving force of 5 tons weight, and the resistances are \(v^2/1000\) tons weight, where \(v\) is the velocity in miles per hour. If the train is started from rest up an incline of 1 in 300, find the velocity at the end of a mile. \[ [\log_{10}e = 0.4343.] \]
A light rod of length \(a\) has at one end a particle, and at the other end a smooth ring of equal mass, which is free to move along a straight horizontal wire. The rod is held with the particle in contact with the wire, and is then released. Shew that, when the rod has turned through an angle \(\theta\), the velocity of the particle is \[ \left\{ag\sin\theta\frac{4-3\sin^2\theta}{2-\sin^2\theta}\right\}^{\frac{1}{2}}. \]
Evaluate \[ \int \frac{dx}{\tan x + c}. \] Shew that \[ \int_0^\pi \frac{(x-1)^4}{(x+1)^5} dx = \frac{1}{5} \quad \text{and} \quad \int_0^{\pi/2} \sqrt{\tan x} dx = \frac{\pi}{\sqrt{2}}. \]
Prove that conics through four fixed points on a circle with centre \(O\) have their axes parallel to fixed directions, and that the locus of the centres of such conics is a rectangular hyperbola \(H\) whose asymptotes are parallel to the axes of the conics. \par Shew further that \(H\) meets each conic \(S\) of the pencil in the feet of the four normals from \(O\) to \(S\).
Two equal particles are joined by a light inextensible string of length \(\pi a/2\) and rest symmetrically on the surface of a smooth circular cylinder of radius \(a\), the axis being horizontal. If the particles are slightly disturbed, show that one of the particles will leave the surface at a height \[ \frac{1}{5}(2\sqrt{2}-\sqrt{3})a \] above the axis of the cylinder. The motion takes place in one plane.
A box of mass \(M\) rests on a rough horizontal table and from the centre of the lid of the box there hangs a pendulum of length \(l\) whose bob has mass \(m\). The pendulum vibrates through a right angle on either side of the vertical. Assuming that the box does not tilt, shew that it does not slide on the table if the coefficient of friction between box and table is greater than \[ \frac{3m}{2\sqrt{M(M+3m)}}. \] If the table had been smooth, shew that the box would have oscillated through a distance \[ \frac{2ml}{M+m}. \]
Obtain a formula of reduction for \(\int \frac{\sin^m\theta}{\cos^n\theta}d\theta\), where \(m\) and \(n\) are positive and greater than 2. \par Shew that \[ \int_0^{\pi/4} \frac{\sin^6\theta}{\cos^8\theta}d\theta = \frac{5\pi}{8} - \frac{23}{12}, \] and evaluate \[ \int_0^{\pi/4} \frac{\sin^8\theta}{\cos^5\theta}d\theta. \]
\(A, B, C, P\) are four points in a plane. The line through \(A\) harmonically conjugate to \(AP\) with respect to the line pair \(AB, AC\) meets \(BC\) in \(L\); \(M\) and \(N\) are similarly defined on \(CA\) and \(AB\). Shew that \(L, M, N\) lie on a line \(p\) (the harmonic polar of \(P\) with respect to the triangle \(ABC\)). \par \(P\) moves on a conic \(S\) through \(A, B, C\). Prove that its harmonic polar passes through a fixed point \(O\), whose harmonic polar is its polar with respect to \(S\). \par \(A', B', C'\) are the points in which \(S\) is met again by the lines joining the vertices of the triangle \(ABC\) to the poles (with respect to \(S\)) of the opposite sides. Shew that the harmonic polar of any point of \(S\) with respect to the triangle \(A'B'C'\) also passes through \(O\).
Two particles of mass \(m\) and \(2m\) are hanging in equilibrium attached to the end of a light elastic string, of unstretched length \(l\) and modulus of elasticity \(mgl/a\). The particle of mass \(2m\) is suddenly removed. Show that the other particle will come to rest again after a time \[ \left(\frac{2\pi}{3}+\sqrt{3}\right)\sqrt{\frac{a}{g}}. \]
Two particles of mass \(m\) and \(M\) are connected by a light inextensible string of length \(2l\) which passes through a smooth hole in a smooth horizontal table on which the mass \(M\) moves while \(m\) hangs vertically. Initially \(M\) is at rest, at a distance \(l\) from the hole, and it is then projected with velocity \(V\) at right angles to the string. Shew that if \(3MV^2 > 8mgl\), \(m\) will reach the hole with velocity \[ \sqrt{\frac{3MV^2-8mgl}{4(M+m)}}. \]