The polar coordinates at time \(t\) of a particle moving in a plane are \(r\) and \(\theta\). Shew that its velocity \(v\) is given by \[ v^2 = \left(\frac{dr}{dt}\right)^2 + \left(r\frac{d\theta}{dt}\right)^2. \] A small ring of mass \(m\) is free to slide along a smooth parabolic wire in a horizontal plane and it is attached to one end of an inextensible string. In the same horizontal plane and in the neighbourhood of the focus there is a table, and the string is threaded through a smooth hole in the table at the focus. At the other end of the string hangs a bead of mass \(M\). The ring is held at one end of the latus rectum (which is of length \(4a\)) and then released. Shew that it will not come to rest again until it reaches the other end of the latus rectum. Also shew that its angular velocity about the focus is greatest as it crosses the axis of the parabola, and find this maximum value.
A block of mass \(M\) with a plane base is free to slide on a smooth horizontal plane. The block contains a spherical cavity, of radius \(a\), whose surface is smooth. A particle of mass \(m\) slides on the surface of the cavity starting from rest at the level of the centre. Shew that when the radius to the particle makes an angle \(\psi\) with the vertical, the velocity of the particle relative to the block is \[ \{2ga \cos\psi (M+m)/(M+m\sin^2\psi)\}^{\frac{1}{2}}, \] assuming the motion of every point to be parallel to the same vertical plane.
Given an ellipse \(a^2y^2 + b^2x^2 - a^2b^2 = 0\), denote by \(N\) the length of the part of the normal at a point \((x, y)\) between the curve and the \(X\)-axis. Prove that the radius of curvature \(\rho\) at the same point is given by the formula \[ \rho = \frac{N^3 a^2}{b^4}. \]
Referred to rectangular axes, the equations of a curve are given in the parametric form \[ x = at + bt^2, \] \[ y = ct + dt^2, \] where \(a, b, c, d\) are constants such that \(ad-bc\) is not zero. Shew that the curve is a parabola and that the chord joining the points whose parameters are \(t_1\) and \(t_2\) is given by the equation \[ \begin{vmatrix} x & y & t_1 t_2 \\ a & c & t_1+t_2 \\ b & d & -1 \end{vmatrix} = 0. \] Further, if the tangents at these two points are at right angles, shew that the chord passes always through the point \((x_0, y_0)\), where \[ \frac{cx_0 - ay_0}{c^2+a^2} = \frac{dx_0-by_0}{2(ab+cd)} = \frac{bc-ad}{4(b^2+d^2)}. \]
A flat disc, with its plane horizontal, is spinning in frictionless bearings at an angular velocity \(\omega_1\) about a vertical axis through its centre, its moment of inertia about that axis being \(I\). A uniform ring of mass \(m\) and radius \(R\), with its plane horizontal and its centre on the axis of the disc, is lowered on to the latter while spinning in its own plane about its centre with an angular velocity \(\omega_2\) in the opposite direction to \(\omega_1\). If the coefficient of friction between the ring and the disc be \(\mu\), derive an expression for the time during which relative slipping will continue.
Two particles of masses \(m\) and \(m'\) are joined by a light inextensible string of length \(a+b\) and rest on a smooth horizontal plane at points \(A, B\) at distances \(a, b\) from a smooth vertical peg \(O\) round which the string passes so that initially the two portions \(OA, OB\) are at right angles. Shew that if the first particle is projected with velocity \(u\) parallel to \(OB\), its distance \(r\) from \(O\) at time \(t\) is given by \(\dot{r}^2 = a^2 + \frac{m}{m+m'}u^2t^2\) if the string is still in contact with the peg.
Shew that, if \(z = x\phi\left(\frac{y}{x}\right) + \psi\left(\frac{y}{x}\right)\), where \(\phi\) and \(\psi\) are any two functions, then \[ x^2\frac{\partial^2 z}{\partial x^2} + 2xy\frac{\partial^2 z}{\partial x\partial y} + y^2\frac{\partial^2 z}{\partial y^2} = 0. \]
If \(S=0\) and \(p=0\) are the equations of a fixed conic and a fixed line, interpret the equation \[ S + \lambda p^2 = 0, \] where \(\lambda\) is a parameter. Each member of a family of conics touches two fixed lines at fixed points \(A\) and \(B\). Shew that the sides of a given triangle meet the polars of the opposite vertices with respect to any one conic of the family in three points which lie on a straight line, and further that the envelope of this line for all members of the family is a conic which touches the sides of the triangle and also touches the line \(AB\).
An elastic string of natural length \(2c\) has its ends attached to the upper corners of a square picture frame of side \(2c\). The string passes over a rough peg and the frame hangs symmetrically below, each half of the string making an angle \(60^\circ\) with the horizontal. The frame is pulled down through a small distance and then released. Shew that it will oscillate up and down and that the period of the small oscillations is the same as that of a simple pendulum of length \(\displaystyle\frac{4c\sqrt{3}}{7}\).
A uniform rod free to turn about its centre \(O\) rests in a horizontal position. A smooth uniform sphere of mass \(m\) falling vertically strikes the rod directly at a distance \(a\) from \(O\). Prove that the rod turns about \(O\) with angular velocity \[ mua(1+e)/(I+ma^2), \] where \(I\) denotes the moment of inertia of the rod about a perpendicular through \(O\), \(e\) the coefficient of restitution between the sphere and the rod, and \(u\) the velocity of the sphere just before impact. Find also the loss of kinetic energy resulting from the impact.