Prove that, when \(x > -1\), $$\log(1 + x) = \frac{2x}{2 + x} + \frac{2x^3}{3(2 + x)^3} + \frac{2x^5}{5(2 + x)^5} + \cdots$$ Hence, or otherwise, show (i) that \(a_n = [1 + (1/n)]^{n+p}\) decreases as \(n\) increases when \(p > 1\); (ii) that, provided \(n\) is sufficiently large, \(a_n\) increases as \(n\) increases when \(p < 1\); (iii) that \(a_n\) increases throughout as \(n\) increases when \(p \leq 0\).
Suppose that \(u(x)\) and \(v(x)\) are polynomials in \(x\) of degrees \(n\) and \(n-1\) respectively, and that they satisfy identically the relation $$\sqrt{1 - [u(x)]^2} = v(x) \sqrt{1 - x^2}.$$ Prove that \(du/dx = \pm nv(x)\), and deduce that \(u(x)\) satisfies the differential equation $$(1 - x^2) \frac{d^2u}{dx^2} - x \frac{du}{dx} + n^2u = 0.$$ By making the change of variable \(x = \cos t\), or otherwise, deduce that $$u(x) = \pm \cos(n \cos^{-1} x), \quad v(x) = \pm \frac{\sin(n \cos^{-1} x)}{\sqrt{1 - x^2}}.$$
Let $$J_m = \int_0^{\pi} \sin^m \theta \sin(n(\pi - \theta)) d\theta,$$ where \(m\) is a non-negative integer, and \(n\) is not an integer. Obtain a reduction formula connecting \(J_m\) and \(J_{m-2}\), and deduce the expansion $$\cos nx = 1 - \frac{n^2}{2!} \sin^2 x - \frac{n^2(2^2 - n^2)}{4!} \sin^4 x - \frac{n^2(2^2 - n^2)(4^2 - n^2)}{6!} \sin^4 x - \cdots$$ [It may be assumed that $$\frac{(2^2 - n^2)(4^2 - n^2) \cdots (4m^2 - n^2)}{(2m)!} J_{2m} \to 0 \text{ as } m \to \infty.]$$
Two curves \(y = f(x)\) and \(y = g(x)\) are said to have \(n\)th order contact at \(x = x_0\) if $$f(x_0) = g(x_0), f'(x_0) = g'(x_0), \ldots, f^{(n)}(x_0) = g^{(n)}(x_0).$$ Prove that there is a conic having fourth order contact at \(x = 0\) with the curve $$y = ax^2 + bx^3 + cx^4,$$ where \(a\), \(b\) and \(c\) are given constants, and find its equation.
Given four points \(P\), \(Q\), \(R\), \(S\) on a rectangular hyperbola with \(PQ\) perpendicular to \(RS\), prove that each of the four points is the orthocentre of the triangle formed by the other three. Let \(PS\) meet \(QR\) in \(D\), \(QS\) meet \(PR\) in \(E\), \(RS\) meet \(PQ\) in \(F\); prove that $$SP \cdot SD = SQ \cdot SE = SR \cdot SF.$$ If \(A\) is the mid-point of \(PQ\) and \(B\) is the mid-point of \(RS\), prove that the circle on \(AB\) as diameter passes through the centre of the hyperbola.
The conics \(S\) and \(S'\) have the equations (in homogeneous coordinates) $$(y + z)^2 + 2zx = 0, \quad y^2 - zx = 0$$ respectively. The polar line of a given point \(P\) on \(S'\) with respect to \(S\) meets \(S'\) in the points \(Q\) and \(R\). Prove that the triangle \(PQR\) is self-polar with respect to \(S\). Show that there is a fixed conic \(S''\) that is inscribed in the triangle \(PQR\) for every position of \(P\) on \(S'\), and find the equation of \(S''\).
Describe briefly the laws of friction as applied to simple problems in mechanics. A particle of mass \(m_1\) is attached to another of mass \(m_2\) by a smooth, light, rigid rod. The system is placed on a rough inclined plane carrying a smooth pin which passes through the hole in the centre of the rod. Show that equilibrium is possible with the rod at any angle in the plane if $$\mu > \frac{|m_1 - m_2|}{m_1 + m_2} \tan \alpha,$$ where \(\mu\) is the coefficient of friction between the particles and the plane, and \(\alpha\) is the inclination of the plane to the horizontal. Write down the equation of motion for the case when the rod is made to rotate in the inclined plane by the application of a couple \(G\).
A four-wheeled truck runs freely on level ground. The distance between the front and rear axles is \(l\), the centre of gravity of the truck is at distance \(b\) from the vertical plane through the front axle and at height \(h\) above the ground. The moments of inertia of the wheels and axles are negligible. Show that, if the rear wheels become locked, the deceleration of the truck is $$f = \frac{\mu bg}{a + \mu h},$$ where \(\mu\) is the coefficient of friction between the wheels and the ground. Describe the way in which \(f\) depends on the parameters \(\mu\), \(b\) and \(h\), and give qualitative arguments in support of the correctness of the formula. Show that, if the front rather than the rear wheels become locked, the condition for the rear wheels to remain on the ground is \(\mu < b/h\). Find the deceleration when this condition is satisfied. Discuss the case when the front and rear wheels become locked simultaneously. In particular, for \(\mu < b/h\), show that the deceleration exceeds that in the two previous cases.
A smooth, plane, unbounded lamina is kept in rotation with constant angular velocity \(\omega\) about a fixed horizontal axis in the plane of the lamina. When the angle which the lamina makes with the vertical is \(\frac{1}{4}\pi\), and is increasing, a particle is placed gently on the upper surface of the lamina at a point on the axis of rotation. Show that, as long as the particle remains on the surface of the lamina, its distance \(r\) from its original position after a time interval \(t\) is $$\frac{g}{2\sqrt{2\omega^2}} [e^{-\omega t} - \sqrt{2} \cos(\frac{1}{4}\pi + \omega t)].$$ Deduce, by reference to a graph, or otherwise, that the particle remains on the surface of the lamina until the lamina is nearly vertical, and that meanwhile the value of \(r\) passes through a maximum.
A smooth hollow circular cylinder of mass \(M\) and radius \(a\) rests on a horizontal plane. A particle of mass \(m\) is released from rest at a point \(P\) on the inner surface of the cylinder, the line joining the middle point of the axis of the cylinder to \(P\) being normal to the axis and making the acute angle \(\alpha\) with the downward vertical. Show that the motion is periodic with period $$2 \sqrt{\frac{2a}{g}} \int_0^{\alpha} \sqrt{\frac{1 - \frac{m}{m + M} \cos^2 \theta}{1 - \cos \theta - \cos \alpha}} d\theta.$$ Hence, or otherwise, show that when \(\alpha\) is small the period is approximately $$2\pi \sqrt{\frac{a}{g} \frac{M}{m + M}}.$$