A small sphere is projected with velocity \(V\) in a vertical plane from a point \(O\) and subsequently strikes a plane through \(O\), with the line of greatest slope in the vertical plane and inclined at angle \(\alpha\) to the horizontal. Find the range \(x\) up the plane when \(\theta\) is the inclination of the velocity of projection to the line of greatest slope of the plane. If \(e\) is the coefficient of restitution of the impact of the sphere and the plane, prove that if \(2\tan\theta=(1-e)\cot\alpha\), the distances between successive impacts are in the decreasing ratio \(e^2\).
A bead of mass \(m\) moves on a smooth wire bent in the form of a circle of radius \(a\) which is held fixed in a vertical plane. An elastic string of natural length \(\mu a\) (\(0<\mu<2\)) has one end attached to the bead and the other end to the highest point of the wire. When the bead is released from rest at the position where the string is just taut, it is found that it comes to rest again at the lowest point of the wire. Prove that \(\mu\) must be the positive root of the equation \[ x^2+x(2+n)-2n=0, \] where \(n\) is the ratio of the modulus of the string to the weight of the bead.
A uniform chain of total mass \(m\) and length \(l\) is released from rest when held vertically with its lower end just touching the bottom of the interior of a bucket of mass \(M\). When half the chain has fallen into the bucket, the bucket itself is released and allowed to fall. It may be assumed that the impact of the chain and bucket is completely inelastic. In the subsequent motion, \(x\) is the distance the bucket has fallen and \(y\) is the length of chain remaining above the bucket. Show that the momentum is given by \[ (M+m)\dot{x} - my\dot{y}/l. \] Show further that the velocity of the chain at the same moment relative to the bucket is \[ (gl)^{\frac{1}{2}} \left(M+\frac{m}{2}\right) / \left(M+m\left(1-\frac{y}{l}\right)\right). \]
A compound pendulum has a detachable rider which it can shed as it passes through its equilibrium position in one sense and can resume as it passes through it in the reverse sense. The equilibrium position is the same with or without the rider attached. The moment of inertia about the horizontal axis from which the pendulum swings is \(I_1\) with the rider attached and \(I_2\) without it. The pendulum carrying the rider is released from rest at an angular displacement \(\alpha\) from the equilibrium position and first sheds and then resumes the rider as it passes and repasses the equilibrium position, and instantaneously comes to rest again on the original side at an angular displacement of \(\beta\). Prove that \[ I_1^2(1-\cos\beta) = I_2^2(1-\cos\alpha). \]
A smooth rigid wire bent in the form of a circle of radius \(a\) and centre \(C\) is constrained to rotate in its own plane (horizontal) with constant angular velocity \(\omega\) about a point \(A\) of its circumference. A bead \(P\) can move on the wire and \(\theta\) is the angle \(ACP\) measured from \(CA\) in the same sense as \(\omega\). By considering the acceleration of the bead along the tangent to the wire at \(P\), show that \[ \ddot{\theta} = \omega^2\sin\theta. \] If \(\dot{\theta}=2\omega\) when the line \(ACP\) is a diameter, prove that in the subsequent motion \[ \dot{\theta} = 2\omega \sin\frac{\theta}{2}. \]
A sphere of radius \(b\) resting on the top of a fixed rough hemisphere of radius \(a\) with horizontal base is displaced slightly and rolls down the hemisphere without slipping. Show that the sphere will leave the surface of the hemisphere when the line joining the centre of the sphere to that of the hemisphere is inclined to the upward vertical at angle \(\theta\), where \[ \cos\theta = \frac{2}{7}\left(3+\frac{\kappa^2}{b^2}\right), \] \(\kappa\) being the radius of gyration of the sphere about a diameter.
The nine numbers \(a_{ij}\) (\(i,j=1, 2, 3\)) satisfy the equations \[ a_{i1} a_{j1} + a_{i2} a_{j2} + a_{i3} a_{j3} = \delta_{ij} \quad (i, j = 1, 2, 3), \] where \(\delta_{ij}=0\) if \(i \neq j\) but \(\delta_{ii}=1\) if \(i=j\). Show that they also satisfy the equations \[ a_{1i} a_{1j} + a_{2i} a_{2j} + a_{3i} a_{3j} = \delta_{ij} \quad (i, j=1, 2, 3). \] Prove also that \(a_{22} a_{33} - a_{23} a_{32} = \pm a_{11}\).
A circle is divided into \(n\) sectors by drawing \(n\) radii. Show that the number of ways of colouring the \(n\) sectors using three given colours so that neighbouring sectors are coloured differently is \[ 2^n + (-1)^n 2. \] (When \(n\) is even, all three colours need not be used.)
Given three collinear points \(A, B, C\) in a plane, explain how to construct the harmonic conjugate of \(C\) with respect to \(A\) and \(B\), using the ruler alone. If \(XYZ\) is the diagonal point triangle of the quadrangle \(ABCD\), prove that \(X\) is the pole of \(YZ\) with respect to any conic \(S\) through \(ABCD\). If a fifth point \(E\) on \(S\) is given, show how to construct with a ruler at least one more point on \(S\).
Given two points \(A, B\) on a conic \(S\), show that there is a unique conic \(S'\) touching \(S\) at \(A\) and \(B\) and such that there exists a triangle inscribed in \(S\) whose sides touch \(S'\). If \(XYZ\) is such a triangle and if \(YZ\) meets \(AB\) in \(U\) and touches \(S'\) in \(V\), show that \(U, V\) separate \(Y, Z\) harmonically.