Problems

A smooth wedge of mass \(M\) and angle \(\alpha\) lies on a horizontal table, and a particle of mass \(m\) is allowed to slide from rest down the inclined face of the wedge. If in the ensuing motion the wedge slides without rotating, show that its acceleration is \[ (mg \cos \alpha \sin \alpha)/(M + m \sin^2 \alpha). \] Find also the reaction between the particle and the wedge, and that between the wedge and the plane.

Two conics touch at \(A\) and intersect at \(B\) and \(C\). Prove that the point \(A\), the middle points of \(BC, CA\) and \(AB\), and the centres of the conics lie on a conic.

If \[ I_{m, n} = \int \cos^m x \cos nx dx, \] prove that \[ (m+n) I_{m,n} = \cos^m x \sin nx + m I_{m-1, n-1}. \] Find \[ \int_0^{\pi/2} \cos^m x \cos nx dx, \] when \(m\) and \(n\) are integers such that \(0 < m < n\).

Define the angular velocity of a rigid lamina moving in its own plane, and prove that, in general, just one point of the lamina is instantaneously at rest. \par Show that, if two such rotations of angular velocities \(\omega_1, \omega_2\) about points \(P_1, P_2\) respectively are superposed, the resultant motion is a rotation of angular velocity \(\omega_1+\omega_2\) about \(G\), the centre of mass of a mass \(\omega_1\) at \(P_1\) and a mass \(\omega_2\) at \(P_2\). \par State and prove the generalisation of this theorem for the case of any number of superposed rotations.

A particle is projected under gravity from a point of an inclined plane in a direction that lies in a vertical plane through a line of greatest slope. Prove that the time taken by the particle to attain its greatest distance from the plane is one-half of the time of flight. \par If the particle can be projected with an assigned initial speed so as to pass through a given point of the plane, show that in general there are two directions of projection. Show also that the product of the times of flight is \(2R/g\), where \(R\) is the range.

Find the coordinates of the mirror image of the point \((h,k)\) in the line \[ lx+my+n=0. \] Prove that the rectangular hyperbolas \begin{align*} xy &= c^2, \\ xy - 2c(x+y)+3c^2 &= 0 \end{align*} touch each other and that each is the mirror image of the other in the common tangent.

Sketch the graph of the function \[ y = e^{-a(x+b/x^2)}, \] where \(a\) and \(b\) are both positive. Prove that there are always at least two points of inflexion. Find the abscissae of the points of inflexion when \(a=10/27\), \(b=5\).

Particles \(P_1, P_2, \dots, P_n\) of the same mass are placed on a smooth horizontal table at the vertices of a regular polygon of \(n\) sides each of length \(a\), and are joined by light inextensible strings \(P_1P_2, P_2P_3, \dots, P_{n-1}P_n\) of length \(a\). (There is no string \(P_n P_1\).) The system is set in motion by a given impulse \(T_1\) applied to \(P_1\) along the line \(P_1P_n\). Denoting the impulses in the strings by \(T_2, T_3, \dots, T_n\) respectively, prove that \[ T_r - 2T_{r+1} \sec \frac{2\pi}{n} + T_{r+2} = 0, \] where \(r=1, 2, \dots, n-2\). (It is helpful to consider the component velocities of \(P_r\) and \(P_{r+1}\) along \(P_rP_{r+1}\).) \par Prove that these equations have a solution of the form \[ T_r = A\alpha^r + B\beta^r, \] where \(A, B, \alpha, \beta\) are constants; and, without carrying out the calculations, explain how to determine these constants.

A small ring slides on a smooth circular wire of radius \(a\) fixed in a vertical plane, and is connected to the highest point of the wire by a light spring of natural length \(a\). Prove that, in any motion in which the spring remains taut, the energy is constant. \par If the ring is in equilibrium at one end of the horizontal diameter, and is then slightly disturbed, prove that it will oscillate with period \(2\pi/n\), where \[ n^2 = \frac{\sqrt{2}+1}{2}\frac{g}{a}. \]

Find the length and direction of the major axis of the ellipse \[ 24x^2 + 8xy + 18y^2 = 1. \] Prove that this ellipse lies entirely inside the ellipse \[ 23x^2 + y^2 = 1. \] % Question 10 is cut off in the image