Problems

A tray of mass \(m\) hangs freely at the lower end of a spring for which the modulus is \(\lambda\). The upper end of the spring is held fixed and a mass \(M\) falls from a height \(h\) on to the tray, which is at rest. During the resulting motion the mass \(M\) remains on the tray. Shew that this motion is simple harmonic, find the amplitude \(a\) of the swing, and shew that the time that elapses after the impact before the tray is next at the same height is \[ \mu\{\pi + 2\sin^{-1}(Mg/a\lambda)\}, \] where \[ \mu^2 = (M+m)/\lambda. \]

Two conics \(S, S'\) have three-point contact at \(P\), and intersect again at \(Q\). \(PT\) is the tangent at \(P\), and \(A, A'\) are the points of contact of the other common tangent. Prove that the lines \(PT, PQ\) are harmonically conjugate with respect to the lines \(PA, PA'\).

Find the limiting values as \(x\) tends to \(0\) of

1. [(i)] \(\displaystyle \frac{\log(\cos x)}{x \sin x}\),
2. [(ii)] \(\displaystyle \frac{1}{e^{x^2}-1} \int_0^x \sin t^2 dt\),
3. [(iii)] \(\displaystyle \frac{1 - \sec(1-\cos x)}{1 - \cos(1-\sec x)}\).

A uniform rod has its upper end attached to and free to slide along a smooth horizontal rail. The rod is held in contact with the rail and released. When it is inclined at an angle \(\pi/4\) to the vertical, its upper end is suddenly fixed. Shew that the greatest angular displacement from the vertical during the subsequent motion is \[ \cos^{-1} \frac{3\sqrt{2}}{16}. \]

A uniform solid circular cylinder of mass \(m\) and radius \(a\) is rolled with its axis horizontal up a rough inclined plane by means of a constant couple \(L\). Shew that, for this to be possible, the coefficient of friction must be greater than \[ \frac{1}{3} \tan\theta + \frac{2}{3} \frac{L \sec\theta}{mag} \] where \(\theta\) is the inclination of the plane to the horizontal.

Two triangles \(ABC, A'B'C'\) in a plane are such that \(AA', BB', CC'\) are concurrent in a point \(O\). \(BC, B'C'\) meet in \(L\); \(CA, C'A'\) in \(M\), and \(AB, A'B'\) in \(N\). Prove that \(L, M, N\) are collinear. Shew further that there exists a unique conic \(S\) with respect to which the triangles reciprocate into each other, and that the polar of \(O\) with respect to \(S\) is the line \(LMN\).

A function \(f(x, y)\), when expressed in terms of the new variables \(u, v\), defined by the equations \[ x = \tfrac{1}{2}(u+v), \quad y^2 = uv, \] becomes \(g(u, v)\). Prove that \[ \frac{\partial^2 g}{\partial u \partial v} = \frac{1}{4} \left( x^2 \frac{\partial^2f}{\partial x^2} + 2x \frac{\partial^2f}{\partial x \partial y} + \frac{\partial^2f}{\partial y^2} + \frac{1}{y} \frac{\partial f}{\partial y} \right). \]

State one set of conditions for the equilibrium of forces acting in a plane upon a rigid body. The ends A and C of two uniform unequal rods AB, BC which are smoothly jointed at B are free to move on a fixed smooth horizontal wire. Shew that the smaller rod, in a position of equilibrium, is vertical; and that the reaction at the wire for the larger rod is independent of the weight of the smaller.

A thin uniform rod is bent at one end to form a walking-stick with a semicircular handle. The straight portion AB is of length \(2l\), and the curved portion BC is of radius \(a\) (less than \(l\)). The straight line joining the ends B and C of the semicircular portion is perpendicular to AB. The stick hangs from a horizontal table, supported at the end C. Prove that the straight part makes an angle \(\psi\) with the vertical, where \[ \tan\psi = a(\pi a + 4l)/2(l^2 - a^2). \]

A heavy elastic string whose weight per unit length when unstretched is \(w\), and whose modulus of elasticity is \(\lambda\), is hung freely from one end A, and supports a weight \(W\) at the other. Prove that the natural length \(l\) is increased to \[ l + \frac{wl^2}{2\lambda} + \frac{Wl}{\lambda}. \] If a rigid horizontal plane is placed underneath \(W\), partly supporting it, at a distance \(h\) from A, prove that the reaction at the plane is \[ W + \lambda + \frac{wl}{2} - \frac{\lambda h}{l}, \] where \[ l + \frac{wl^2}{2\lambda} \le h \le l + \frac{wl^2}{2\lambda} + \frac{Wl}{\lambda}. \]