Problems

Find all the values of \(x\), \(y\) and \(z\) which satisfy the equations \begin{align} -y + z &= u,\\ x - z &= v,\\ -x + y &= w, \end{align} where \begin{align} v - 2w &= a,\\ -u + 3w &= b,\\ 2u - 3v &= c. \end{align}

Show that \[ (-1)^n e^{z^2} \frac{d^n e^{-z^2}}{dz^n} \] is a polynomial of degree \(n\) in \(z\). Call this polynomial \(H_n(z)\), and show, in any order, that

1. [(i)] \(\frac{d}{dz} H_n(z) = 2nH_{n-1}(z)\);
2. [(ii)] \(H_{n+1}(z) - 2zH_n(z) + 2nH_{n-1}(z) = 0\);
3. [(iii)] \(\frac{d^2H_n(z)}{dz^2} - 2z\frac{dH_n(z)}{dz} + 2nH_n(z) = 0\).

Also show that \(H_n(z)\) has \(n\) distinct zeros.

Show that, if \(x > 0\), \(y > 0\), and \(x + y\),

1. [(i)] \(xx^r(x-y) > x^r - y^r > ry^{r-1}(x-y)\), \((r > 1)\)
2. [(ii)] \(rx^{r-1}(x-y) < x^r - y^r < ry^{r-1}(x-y)\), \((0 < r < 1)\).

Deduce that \[ \frac{N^{r+1} + r}{r+1} < \sum_{1}^{N} n^r < \frac{(N+1)^{r+1} - 1}{r+1}, \quad (0 < r). \]

Show that the product of two involutions is another involution if and only if the double points of the first involution are mates in the second involution. Also, show that the product of three involutions is another involution if and only if the pairs of double points belong to an involution. Show that if \(I_1\), \(I_2\), \(I_3\), \(I_4\) \(I_5\) and \(I_1I_2I_3\) are involutions, then so are \(I_2I_1\), \(I_3I_1I_3\) and \(I_1I_2\). \(I_4\), \(I_5\), \(\ldots\), \(I_n\) are further involutions such that \(I_1I_2I_n\), \(\ldots\), \(I_1I_2I_n\) are involutions; show that \(I_1I_2I_k\) is an involution, where \(4 \leq i < j < k \leq n\). (The mate by \(I'\) of the mate by \(I\) of \(P\) is the transform of \(P\) by the product \(I'I\).)

Write down

1. [(i)] the equation of a conic having double contact with a conic \(S\) at two distinct points \(A\), \(B\) of \(S\);
2. [(ii)] the equation of a conic having three-point contact with a conic \(S\) at a point \(A\) of \(S\) and passing through another point \(B\) of \(S\);
3. [(iii)] the equation of a conic having four-point contact with a conic \(S\) at a point \(A\) of \(S\).

Two conics \(S_1\), \(S_2\) pass through two distinct points \(A\), \(B\) of a conic \(S\) and have three-point contact with \(S\) at \(A\), \(B\) respectively. The line joining the pole of \(AB\) with respect to \(S_1\) to the pole of \(AB\) with respect to \(S_2\) meets the line \(AB\) at \(D\). Prove that if \(C\) is the pole of \(AB\) with respect to \(S\) then \(CD\) and the common chord of \(S_1\), \(S_2\) other than the line \(AB\) harmonically separate the lines \(CA\), \(CB\).

A point \(A\) is fixed above a rough plane, which is inclined at an angle \(\alpha\) to the horizontal. A uniform rod has one end freely jointed at \(A\), and rests with its other end \(B\) on the plane. The acute angle \(\beta\) between the rod and the normal to the plane is greater than \(\alpha\). Show that every position of the rod is one of equilibrium provided that the coefficient of friction exceeds \[ \frac{\sin\alpha\sin\beta}{\sqrt{(\sin^2\beta - \sin^2\alpha)}}. \] [It is to be assumed that the direction of the frictional force is at right angles to the rod.]

A uniform rod of length \(2a\) is smoothly hinged at one end to a fixed point \(A\) of a horizontal axis in such a way that it must lie in the plane through \(A\) perpendicular to the axis. The rod is kept in rotation with constant angular velocity \(\omega\) about the vertical through \(A\). Show that, if \(\omega\) is sufficiently large, there is a position of relative equilibrium in which the rod makes an angle \(\alpha\) with the vertical, given by \(\cos\alpha = 3g/(4\omega^2)\). Show also that the equilibrium is stable, and that the period of small oscillations about the equilibrium position is \(2\pi/(\omega\sin\alpha)\). Derive corresponding results for the case in which the rod is non-uniform.

Two identical toothed wheels \(W_1\) and \(W_2\), in a common vertical plane, can spin about smooth axes through their centres. Each wheel has radius \(a\) and moment of inertia \(I\) about its axle. Initially the axles are sufficiently far apart for the wheels not to be in contact, and \(W_1\) has angular velocity \(\Omega\), whilst \(W_2\) is at rest. The axles are then brought closer together, so that the wheels engage. Find the subsequent angular velocity of the wheels and the resulting loss of energy. The procedure is repeated, with the difference that the wheel \(W_2\), though still initially at rest, now has wrapped round its perimeter part of a light chain, the remainder of which hangs vertically from the level of the axle and is attached to a mass \(m\). When the wheels engage, the rotation of \(W_2\) winds up the chain and the attached mass, without the latter reaching the rim of \(W_2\). Show that the time which elapses before the system comes to rest again is \[ I\Omega/(mga), \] and that the height through which the mass has then been raised is \[ \frac{I^2\Omega^2}{2mg(2I + ma^2)}. \]

A ball, of radius \(a\) and radius of gyration \(k\) about a diameter, lands with back spin on a rough plane inclined at an angle \(\alpha\) to the horizontal. Immediately after the impact, which is inelastic, the centre of the ball has velocity \(V\) in the upwards direction of a line of greatest slope, and the ball is turning, in the vertical plane through this line, with angular velocity \(\Omega\). Show that the centre of the ball comes to rest before slipping ceases if \[ \Omega > \frac{\mu\cos\alpha}{\mu\cos\alpha + \sin\alpha} \cdot \frac{aV}{k^2}, \] where \(\mu\) is the coefficient of friction; and on the assumption that this is the case, find the distance travelled up the slope before the centre of the ball comes to rest.

A particle of unit mass is attached to one end of a light spring, the other end of which is fixed to a distant point on a smooth horizontal plane. The modulus of the spring is \(n^2\) times its natural length. The particle moves on the plane in a straight line under the joint influence of the spring, a retarding force of amount \(2k\) times the speed of the particle, and a force \(F\cos(\Omega t)\), where \(t\) is the time and \(F\) and \(\Omega\) are constants. At \(t = 0\) the particle is at rest, and the spring is neither stretched nor compressed. Determine the subsequent motion of the particle, distinguishing between the cases in which \(n^2\) is greater than, equal to, or less than \(k^2\).