Problems

A particle slides, from rest at a depth \(r/2\) below the highest point, down the outside of a smooth sphere of radius \(r\); prove that it leaves the sphere at a height \(r/3\) above the centre. Shew further that when the particle is at a distance \(r\sqrt{2}\) from the vertical diameter of the sphere it is at a depth \(4r\) below the centre.

Two particles \(A, B\) of masses \(2m\) and \(m\) respectively are connected by a light rod and lie on a smooth horizontal table. If the mass \(A\) is struck a blow in a direction \(\tan^{-1}\frac{1}{2}\) with \(AB\), prove that the initial velocity of \(A\) is \(\sqrt{5}\) times that of \(B\).

Explain the geometrical method known as generalization by projection, and generalize the following results:

1. [(i)] the radical axes of three circles taken in pairs are concurrent;
2. [(ii)] tangents to a parabola from a point on the directrix are perpendicular;
3. [(iii)] confocal conics cut orthogonally.

If \(\xi = ax+hy+g\) and \(\eta=hx+by+f\), prove that for the conic \[ S \equiv ax^2+by^2+c+2fy+2gx+2hxy = 0 \]

1. [(i)] the asymptotes have equation \(b\xi^2 - 2h\xi\eta + a\eta^2 = 0\);
2. [(ii)] the axes have equation \(h(\xi^2-\eta^2) - (a-b)\xi\eta = 0\);
3. [(iii)] the director circle has equation \((a+b)S - (\xi^2+\eta^2) = 0\);
4. [(iv)] the directrices are given by \((ab-h^2)S + \lambda(\xi^2+\eta^2) = 0\), where \(\lambda\) is a root of \(\lambda^2+(a+b)\lambda+ab-h^2=0\).

If \(\alpha\) is a first approximation to a root of an equation \(f(x)=0\), shew that \(\alpha - \dfrac{f(\alpha)}{f'(\alpha)}\) is likely to be a better approximation. Discuss the conditions with reference to \(f'(x), f''(x)\) which tend to vitiate this result. Explain the geometrical significance of your arguments. Apply the method to determine, correct to three places of decimals, the root of \[ x^3 - 8x - 60 = 0 \] which is nearly equal to 3.

Shew that, if \(A, B\) are two polynomials having no common factor, and of degrees \(a, b\) respectively, and if \(P\) is any polynomial of degree \(p\), then polynomials \(X, Y\) can be found such that \[ AX+BY=P. \] Shew that, if \(p < a+b\), then \(X\) and \(Y\) are uniquely determined if we restrict them to have degrees less than \(b, a\) respectively. Explain and illustrate the use of this result in the theory of Partial Fractions.

Defining \(\cos x\) and \(\sin x\) as solutions of the differential equation \(\dfrac{d^2y}{dx^2} + y = 0\) with suitable given values of \(y\) and \(\dfrac{dy}{dx}\) at \(x=0\), and defining \(\dfrac{\pi}{2}\) as the least root of the equation \(\cos x = 0\), obtain, without the use of infinite series, the principal properties of \(\cos x\) and \(\sin x\).

Similar rectangular slabs, \(n\) in number, are placed in a pile, so that at one end each projects beyond the one below it. The \(r\)th slab from the bottom overhangs by an amount \(d_r\). The length of each slab is \(2l\), and its weight \(w\). A load \(W\) is placed on the projecting end of the uppermost slab. The distances \(d_r\) are adjusted so that the pile is on the verge of overturning at every interface. Shew that the distances \(d_r\) increase with \(r\) and are in harmonic progression.

A uniform rod \(AB\) of length \(l\) is constrained without friction so that \(A\) moves on the circumference and \(B\) on the vertical diameter (not produced) of a circle in a vertical plane. The radius of the circle is \(a\). Find the positions of equilibrium, given that \(l\) is between \(\frac{1}{2}a\) and \(a\), and discuss separately the stability of the upper and lower positions. What happens if \(l\) is less than \(\frac{1}{2}a\)?

A smooth wedge of mass \(M\) and angle \(\alpha\) is free to slide on a horizontal plane. A small perfectly elastic ball of mass \(m\) bounces on the wedge, the motion being in a vertical plane of symmetry. Shew that impacts occur after successive equal intervals of time. Immediately before an impact the component velocity of the ball perpendicular to the face of the wedge is \(u_1\), and the velocity of the wedge is \(v_1\). Shew that the corresponding velocities \(u_2, v_2\) immediately before the next impact are \begin{align*} u_2 &= \frac{1}{(M+m\sin^2\alpha)} \{(M+3m\sin^2\alpha)u_1 - 2mv_1\sin\alpha\}, \\ v_2 &= \frac{1}{(M+m\sin^2\alpha)} \{2mu_1\sin\alpha + (M-m\sin^2\alpha)v_1\}. \end{align*}