A particle of mass \(m\) is suspended from a fixed support by a light elastic string. When the mass is in equilibrium the extension of the string is \(c\). Show that, when the extension of the string is \(x\), the energy stored in the string is \(\frac{1}{2}m\omega^2 x^2\), where \(\omega^2=g/c\), and prove that, for vertical oscillations (in which the string does not become slack) about the position of equilibrium, the length of the equivalent simple pendulum is \(c\). If the mass is pulled vertically downwards a distance \(kc\) (\(k>1\)) below the position of equilibrium and released from rest, show that the mass first reaches the highest point of its path in a time \[ \{\pi + \text{cosec}^{-1} k + (k^2-1)^{\frac{1}{2}}\}/\omega. \] It may be assumed that \(k\) is such that the mass does not rise as high as the support.
A rocket is travelling vertically upwards. Its initial mass is \(M\), and a mass of gas \(q\) per unit time is discharged at a constant rate vertically downwards with constant velocity \(V\) relative to the rocket. By considering the change of momentum in a small interval of time \(dt\), show that the equation of motion is \[ m\frac{dv}{dt} = qV-mg, \] where \(m\) is the mass of the rocket and \(v\) its velocity at a time \(t\). Deduce that, if the rocket starts from rest at ground level at \(t=0\), \[ v=V\log\{M/(M-qt)\}-gt. \]
A small bead, of mass \(m\), slides on a smooth wire bent in the form of a parabola, of which the plane is vertical and the vertex is at the lowest point. If the bead is released from rest at one end of the latus rectum, find the reaction of the wire on the bead when the bead is at the vertex.
A plane lamina is moving in its own plane. Show that at any instant the lamina is in general rotating about a point (the "instantaneous centre"). One end \(A\) of a straight connecting-rod \(AB\), of length \(l\), is made to describe a circle, of centre \(O\) and radius \(a\) (\(< l\)); the other end \(B\) is constrained to remain on a straight line \(Ox\) in the plane of the circle. Show in a diagram the position of the instantaneous centre \(I\) of \(AB\) for a general position of \(A\). If \((x,y)\) are the coordinates of \(I\) relative to rectangular axes \(Ox, Oy\) in the plane of the circle, show that \[ (x^2+y^2)(x^2-l^2+a^2)^2=4a^2x^4. \]
A circular wheel, of radius \(a\), of radius of gyration \(k\) and of mass \(M\), is mounted at one end of a light arm, as shown in Fig.~1, so as to be capable of rotation with its axis horizontal. The other end of the arm is freely pivoted to a fixed support as shown. At an instant when the wheel is spinning with angular velocity \(\Omega\) it is allowed to drop gently on to a plank, of mass \(M'\), which is at rest on a smooth horizontal table; the arm is horizontal. If the coefficient of friction between the wheel and the board is \(\mu\) and the board moves without rotation, show that slipping continues for a time \[ \frac{M'k^2\Omega}{(Mk^2+M'a^2)\mu g}. \] Find the distance moved by the board in this time.
\(ABCDE \dots\) is a closed polygon constructed of light rods \(AB, BC, \dots\) freely jointed at the vertices. It is in equilibrium under forces \(P_A\) applied at \(A\), and so on. A second (closed) polygon \(UVWXY \dots\) is constructed, in a similar manner, so that \(UV\) represents \(P_A\) (in magnitude and direction), \(VW\) represents \(P_B\), and so on. At \(V\) is applied a force represented in magnitude and direction by \(AB\), at \(W\) one represented by \(BC\), and so on all round the polygon. Prove that the second polygon is in equilibrium if and only if the lines of action of the forces \(P_A, P_B, \dots\) acting on the first polygon are concurrent.
Two fixed equally rough planes, intersecting in a horizontal line, are inclined at equal angles \(\theta\) to the vertical. A uniform rod rests between the planes, in a vertical plane at right angles to their line of intersection, and makes an angle \(\alpha\) with the vertical. If the rod is about to slip, show that \(\tan(\theta+\lambda) - \tan(\theta-\lambda) = 2 \cot \alpha\), where \(\lambda\) is the angle of friction. Deduce, or show otherwise, that if \(\alpha = \alpha_0\) in the position of limiting equilibrium then all positions with \(\alpha > \alpha_0\) are positions of equilibrium: and hence that if \(\mu\), the coefficient of friction, exceeds the positive root of the equation \(\mu^2 \sin^2 \theta + \mu \tan \theta - \cos^2 \theta = 0\), then all physically possible positions are positions of equilibrium.
A quadrilateral \(ABCD\) is formed from four uniform rods freely jointed at their ends. The rods \(AB\) and \(AD\) are equal in length and weight, and so also are the rods \(BC\) and \(CD\). The quadrilateral is suspended from \(A\) and a string joins \(A\) and \(C\) so that \(ABC\) is a right angle and \(BAD=2\theta\). Show that the tension in the string is \(w' + (w+w')\sin^2\theta\), where \(w\) is the weight of \(AB\) and \(w'\) is the weight of \(BC\).
If \(M(x)\) is the bending moment at a point distant \(x\) from one end of a thin straight horizontal beam of variable weight \(w(x)\) per unit length, show that, with suitable sign conventions, \(d^2M/dx^2 = w(x)\). A beam is required to satisfy the relation \(M(x)=k[w(x)]^2\), where \(k\) is a given constant. If one end of the beam is clamped and the other is free, obtain a differential equation satisfied by \(M(x)\), and hence show that the weight per unit length at distance \(x\) from the free end must be \(x^2/12k\). What are the dimensions of the constant \(k\), in terms of length, mass and time?
A train is running down a slope inclined at an angle \(\alpha\) to the horizontal, the engine exerting no tractive force. It is loading water at a rate \(p\) from a stream which is flowing at a speed \(u\) in the same direction as the train. Show that, if frictional resistance is neglected, the speed \(v\) of the train at time \(t\) is given by \[ (M_0+pt)v = (M_0t + \tfrac{1}{2}pt^2)g\sin\alpha + put + M_0v_0, \] where \(v_0, M_0\) denote the velocity and complete mass of the train at time \(t=0\).