A train of mass \(M\) is moving with velocity \(V\) when it begins to pick up water at a uniform rate. The power is constant and equal to \(H\). If after time \(t\) a mass \(m\) of water has been picked up, find the velocity and shew that the loss in energy is \[ \frac{m(Ht + MV^2)}{2(m+M)}. \]
Two uniform circular cylinders of the same radius rest on an inclined plane and touch along a generator, their axes being horizontal. All the surfaces are rough, with the same coefficient of friction \(\mu\). Shew that, for equilibrium to be possible, the conditions \[ W_2 > W_1, \quad W_2(2\cot\alpha-1) > W_1 \] must be satisfied, where \(W_1\) is the weight of the lower cylinder, \(W_2\) that of the upper cylinder, and \(\alpha\) is the inclination of the plane to the horizontal. If these conditions are satisfied, find the least value of \(\mu\) which makes equilibrium possible for a given \(\alpha\).
Find the limiting values as \(x\) tends to \(0\) of
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 \(PA, PA'\).
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 \[ \mu^2 = (M+m)/\lambda. \]
A bullet of mass \(m\) is fired horizontally with velocity \(V\) into a block of mass \(M\) which rests on a rough horizontal plane, the coefficient of friction being \(\mu\). The block offers a constant resistance \(F\) to the bullet so long as there is any relative motion. Shew that, if \(F > \mu(M+m)g\), the bullet penetrates a distance \[ \frac{mM}{2(m+M)} \frac{V^2}{(F - \mu mg)} \] into the block, and find the time which elapses before the block comes to rest.
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 \[ \frac{\partial^2 g}{\partial u \partial v} = \frac{1}{4} \left( x^2 \frac{\partial^2f}{\partial x^2} + 2x \frac{\partial^2f}{\partial x \partial y} + \frac{\partial^2f}{\partial y^2} + \frac{1}{y} \frac{\partial f}{\partial y} \right). \]
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 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.
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 \[ \cos^{-1} \frac{3\sqrt{2}}{16}. \]