The ends of a light spring of natural length \(2a\) and modulus \(\lambda\) are fixed at points \(A, B\) on a smooth horizontal table at distance \(4a\) apart, and a particle of mass \(m\) is fixed to the mid-point of the spring. Write down the equation of motion of the particle along the horizontal perpendicular bisector of \(AB\), and, by integrating this equation, prove that energy is conserved in the motion. Prove that, for oscillations of small amplitude, the period is approximately \(2\pi \sqrt{\left(\dfrac{ma}{\lambda}\right)}\).
For each of the following, write down an equation of motion, giving your reasons fully, and deduce an expression for the velocity \(v\) after a time \(t\):
Prove that the maxima of the curve \(y=e^{-kx}\sin px\) (\(k\) and \(p\) being positive constants) all lie on a curve whose equation is \(y=Ae^{-kx}\), and find \(A\) in terms of \(k\) and \(p\). Draw in the same diagram rough sketches of the curves \(y=e^{-kx}\), \(y=-e^{-kx}\) and \(y=e^{-kx}\sin px\) for positive values of \(x\).
\(P\) is a variable point of a conic \(S\), and \(Q\) is the centre of the rectangular hyperbola having four-point contact with \(S\) at \(P\). If \(S\) is a circle, show that as \(P\) varies \(Q\) describes a concentric circle of twice the radius. If \(S\) is a parabola, show that the locus of \(Q\) is an equal parabola with the same axis and directrix as \(S\).
Two gear wheels \(A\) and \(B\), of radii \(a, b\) and moments of inertia \(I, I'\) respectively, are mounted so as to be able to rotate without appreciable friction about their respective axes. The wheels are toothed and run permanently in mesh. A constant torque \(G\) is applied to \(A\) about its axis. Find
A number of equal masses \(m\) are joined by light strings of length \(s\) so that the masses are at the angular points of a regular polygon, of side \(s\), inscribable in a circle of radius \(a\). The polygon lies on a smooth horizontal table and rotates steadily in its own plane with angular velocity \(\omega\) about its centre. Show that the tensions in the strings are \(ma^2\omega^2/s\) in absolute units. By considering the limiting case in which the number of sides tends to infinity, find the tension in a uniform string, of mass \(2\pi a \lambda\), which is in the form of a horizontal circle of radius \(a\) and is rotating in its own plane about its centre with uniform angular velocity \(\omega\). Verify this latter result by introducing centrifugal forces and applying the principle of virtual work to the corresponding statical problem.
Evaluate the integrals \[\int_1^2 \{\sqrt{(2-x)(x-1)}\} \,dx, \quad \int_0^\infty (1+x^2)^2 e^{-x} \,dx.\]
The equations \(S=0\), \(u=0\) and \(v=0\) represent respectively a conic and two straight lines. Interpret the equations \(S+\lambda uv = 0\), \(S+\lambda u^2=0\), where \(\lambda\) is a parameter. Two conics \(S_1\) and \(S_2\) each have double contact with \(S\). Show that two of the common chords of \(S_1\) and \(S_2\) meet at the point of intersection of the chords of contact of \(S_1\) and \(S_2\) with \(S\), and form with them a harmonic pencil.
Explain fully what is meant by the dimensions of a physical quantity. The measure of a certain physical quantity is found to be 1 when pound-foot-second units are used, \(16\) when ounce-foot-second units are used, \(9\) when ounce-inch-second units are used, and \(h\) when ton-mile-hour units are used. Find its dimensions in mass, length and time, and compare the unit of this physical quantity in the ounce-inch-second system with that in the ton-mile-hour system.
A light inextensible string of length \(2l\) is fastened at one end to a fixed point; it carries a mass \(m\) at the mid-point and a mass \(2m\) at the lower end. The system is slightly disturbed from rest so that the masses move in the same vertical plane. If the horizontal displacements of the upper and lower masses are \(x_1\) and \(x_2\) respectively (both being measured in the same direction) show that the equations of motion of the masses are \begin{align*} \frac{d^2 x_1}{dt^2} + n^2 (5x_1 - 2x_2) &= 0, \\ \frac{d^2 x_2}{dt^2} + n^2 (x_2 - x_1) &= 0, \end{align*} where \(n^2=g/l\), and terms of higher order are neglected. Show that a state of oscillation is possible in which the masses execute simple harmonic motions of the same period, with \(x_1 = x_2(\sqrt{6}-2)\), and find the period of oscillation.