Considering a rigid body as made up of a number of particles which obey Newton's laws, prove that the resultant sum of the external forces acting on a rigid massive lamina is equal to the mass of the lamina multiplied by the acceleration of its mass-centre. Prove also that the sum of the moments of the external forces about an axis through the mass-centre and perpendicular to the lamina is equal at any instant to the rate of change of the angular momentum about this axis (in general a moving axis). A light inextensible string \(AB\) has one end fixed at \(A\); the string passes under a free pulley \(C\) and over a pulley \(D\) whose axle is fixed, the parts of the string not in contact with the pulleys being vertical as shown in the diagram. The pulleys \(C, D\) have moments of inertia \(I_1, I_2\) respectively about their axles, and radii \(a_1, a_2\) respectively. A force \(P\) is applied at \(B\) to lift a load attached to the axle of pulley \(C\), the mass of the load and pulley together being \(m\). Assuming that the string does not slip on the pulleys, but that friction at the axles can be neglected, find the acceleration of the load. \vspace{1cm} % Diagram description
Prove, by considering \(\int_a^b (f(x)+\lambda g(x))^2 dx\) for all real \(\lambda\), that \[ \left( \int_a^b f(x)g(x) \,dx \right)^2 \le \left( \int_a^b (f(x))^2 \,dx \right) \left( \int_a^b (g(x))^2 \,dx \right). \] (It may be assumed that, if \(\phi(x) \ge 0\) when \(a \le x \le b\), then \(\int_a^b \phi(x) \,dx \ge 0\).) Prove that \[ \int_0^{\pi/2} \sin^{n+1}\theta \,d\theta \le \frac{1}{2^n n!} \sqrt{\frac{(2n)!}{2}\pi}. \]
Two sequences \(a_0, a_1, \dots\); \(b_0, b_1, \dots\) are connected by the relations \[ a_n = \sum_{r=0}^n \binom{n}{r} b_{n-r}, \quad n=0, 1, \dots \] Prove that \[ b_n = \sum_{r=0}^n (-1)^r \binom{n}{r} a_{n-r}, \quad n=0, 1, \dots \] A chess tournament is arranged among \(2n\) schools, each school sending two players. If each player plays one match, find \(a_n\), the number of ways in which the matches can be arranged if players from the same school may play each other. Let \(b_n\) be the number of ways that the matches can be arranged if players from the same school are barred from playing each other. Show that \[ a_n = \sum_{r=0}^n \binom{n}{r} b_{n-r} \] and hence determine \(b_n\). Evaluate \(b_5\).
If \(f(x)\) is a polynomial in \(x\) of degree 2, and \[ F_n(x) = \frac{d^n}{dx^n} [\{f(x)\}^n], \] show that \[ F'_{n+1}(x) = f(x)F''_n(x) + (n+2)f'(x)F'_n(x) + \frac{1}{2}(n+1)(n+2)f''(x)F_n(x). \] Hence or otherwise obtain the relation \[ f(x)F''_n(x) + f'(x)F'_n(x) - \frac{1}{2}n(n+1)f''(x)F_n(x) = 0. \]
A conic \(S\) is inscribed in a triangle \(ABC\) and a conic \(S'\) touches \(AB\) at \(B\) and \(AC\) at \(C\). Show that by suitable choice of homogeneous co-ordinates their equations can be written in the forms \begin{align*} S &\equiv a^2x^2+b^2y^2+c^2z^2-2bcyz-2cazx-2abxy=0, \\ S' &\equiv yz-x^2=0. \end{align*} If a general point of \(S'\) is expressed parametrically as \((1, t, 1/t)\), show that the equation giving the parameters of the points common to \(S\) and \(S'\) can be written in the form \[ (at^2+\lambda t+c)(at^2+\mu t+c)=0, \] where \(\lambda\) and \(\mu\) are the roots of a certain quadratic equation. Hence or otherwise show that two of the common chords of \(S\) and \(S'\) are such that their poles with respect to \(S\) lie on \(S'\).
A region \(R\) in a Euclidean plane is said to be convex if, for each pair of points \(A, B\) both lying in \(R\), the whole of that part of the line \(AB\) which lies between \(A\) and \(B\) lies in \(R\). Prove that the region consisting of the points \((x,y)\) for which \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-1 < 0\) is convex, and that the region for which \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-1 \ge 0\) is not convex.
A bead of mass \(m\) is free to slide on a smooth circular wire of radius \(a\) which is fixed in a vertical plane. The bead is attached to the highest and lowest points of the wire by two light elastic strings of natural length \(a\) and moduli \(\lambda_1\) and \(\lambda_2\) respectively. Show that the bead will be in equilibrium at a point of the circle with vertical tangent if \[ \lambda_1-\lambda_2 = mg(2+\sqrt{2}). \] Investigate the stability of this equilibrium.
A uniform flexible chain of length \(l\) and weight \(wl\) hangs over a rough horizontal cylinder of radius \(a\) (\(l > a\pi\)) in such a way that the chain lies in a vertical plane perpendicular to the axis of the cylinder. Equal lengths of chain hang on either side of the cylinder and a weight \(W\) is attached to one end. If the chain is on the point of slipping, show that the tension \(T\) at points in contact with the cylinder satisfies the equation \[ \frac{dT}{d\theta} + \mu T = wa(\cos\theta-\mu\sin\theta), \] where \(\theta\) is the angular distance from the point where the chain leaves the cylinder on the side to which the weight \(W\) is attached, and \(\mu\) is the coefficient of friction. Find \(T\) as a function of \(\theta\) from this equation by multiplying both sides by \(e^{\mu\theta}\) and hence determine \(W\).
A particle is projected from a point \(O\) so as to strike a smooth vertical wall which is at a distance \(a\) from \(O\) and, after rebounding from the wall, to pass again through \(O\). Prove that the components \((u,v)\) of the velocity of projection must satisfy the relation \[ uv = \frac{1}{2}ga\left(1+\frac{1}{e}\right), \] where \(e\) is the coefficient of restitution. Hence show that the least possible speed of projection is \[ \sqrt[4]{\frac{ga}{e}\left(1+\frac{1}{e}\right)}. \]
A bead of mass \(m\) slides on a smooth wire in the form of the parabola \(x^2=4ay\), which is fixed with its axis vertical and vertex downwards. The bead is released from rest. Prove that during the subsequent motion the horizontal displacement \(x\) of the bead from the axis satisfies the equation \[ \frac{1}{4a^2}\dot{x}^2(4a^2+x^2) = ag(c^2-x^2), \] where \(c\) is the initial value of \(x\). Find the reaction of the wire on the bead when the bead is at the lowest point of the wire. If \(c\) is small, what is the period of the resulting small oscillation?