A compound pendulum is formed by a lamina of mass \(M\) swinging in its own plane, which is vertical. Given that the distance of the point of suspension \(O\) from the centre of gravity \(G\) is \(h\), and that the moment of inertia about the horizontal axis through \(G\) is \(Mk^2\), find the length \(l\) of the simple equivalent pendulum. Find the relation between \(h\) and \(k\) which will ensure that errors in measuring \(h\) may have the least possible effect on the determination of \(l\). In order to eliminate the error due to sway of the supporting framework as the pendulum swings, two equal pendulums are swung from the same horizontal beam in opposing phase, swinging through equal small angles \(\alpha\). Find, to the first order, the horizontal component of the force exerted by one pendulum on its axis when \(OG\) is inclined at an angle \(\theta\) to the vertical. If the pendulums differ in \(h\) by a small amount \(\delta h\), but do not differ in \(M\) or \(k\), and having been started from their vertical positions with equal and opposite velocities become gradually out of step, obtain a first-order formula for the variation of the residual horizontal force on the beam with time. How is this formula simplified if the relation described above holds between \(h\) and \(k\) for one of the pendulums?
A tennis match is played between two teams, each player playing one or more members of the other team. Further, (i) any two members of the same team have exactly one opponent in common; (ii) no two players belonging to the same team play all the members of the other team between them. Prove that two players in opposite teams who do not play each other have the same number of opponents. Deduce that any two players, whether belonging to the same team or different teams, have the same number of opponents.
A polygon \(P\) has vertices \(A_1, \dots, A_n\) where the coordinates \(x_r, y_r\) of \(A_r\) are both integers for \(r=1, \dots, n\). The number of points inside \(P\) (and not on the sides) whose co-ordinates \(x, y\) are both integers is denoted by \(m(P)\) and the number of points on the sides whose coordinates are both integers is denoted by \(m'(P)\); this including the vertices \(A_1, \dots, A_n\). Prove that the area of \(P\) is given by the formula \[ \text{Area} = m(P) + \tfrac{1}{2}m'(P) - 1 \] in the following cases:
The real number \(a\) is greater than 1 and an approximation \(x\) to the square root of \(a\) is given which is also greater than 1. A new approximation \(y\) to the square root of \(a\) is defined by the formula \[ y = \tfrac{1}{2}\left(x+\frac{a}{x}\right). \] Prove that the square root of \(a\) lies between \(y\) and \(y - \frac{1}{8y}(x^2-a)^2\). By taking \(x=1 \cdot 4\), obtain an approximation to \(\sqrt{2}\) which is correct to 3 places of decimals, proving that this is the case.
Prove that \[ \int_1^a \frac{f(x)}{x} dx = \int_{1/a}^1 \frac{f(1/x)}{x} dx \] where \(a>0\). Evaluate the integral \[ \int_{1/a}^a \frac{(\log x)^2 dx}{x(1+x^n)}. \]
The triangle \(ABC\) is acute-angled; \(P\) is a point that can vary on \(BC\) (but not outside the segment \(BC\)). The mirror images of \(B, P\) in \(AC\) are \(B', P'\) and of \(C, P'\) in \(A B'\) are \(C'', P''\). Find for what position of \(P\) on \(BC\) the distance \(PP''\) is least. The points \(Q, R\) vary on the segments \(AC, AB\); the image of \(R\) in \(AC\) is \(R'\). By comparing \(PQ+QR'+R'P''\) with \(PP''\), or otherwise, determine for any given position of \(P\) the positions of \(Q\) and \(R\) for which the perimeter of the triangle \(PQR\) is least; hence determine the positions of \(P, Q\) and \(R\) for which this perimeter is least. [It may be assumed without proof that, for any position of \(P\) in \(BC\), the segment from \(P\) to \(P''\) cuts \(AC, AB'\) internally in that order.]
A regular dodecahedron is bounded by twelve regular pentagons each with side of unit length. Prove that the obtuse angle between two adjacent faces is \(116\frac{1}{2}^\circ\) approximately and find the radii of the inscribed and circumscribed spheres of the dodecahedron.
A uniform solid cylinder of mass \(m\) and radius \(a\) rolls down a rough plane inclined at an angle \(\alpha\) to the horizontal, the coefficient of friction being \(\mu\). A retarding couple \(G\) is then applied to the cylinder for the purpose of stopping the translational motion as quickly as possible. Show that the cylinder will not be stopped unless \[ \mu > \tan\alpha, \quad G > mga\sin\alpha, \] but that there is no advantage to be gained by applying a couple of greater magnitude than \[ \tfrac{1}{2}mga(3\mu\cos\alpha - \sin\alpha). \]
A smooth and perfectly elastic ball is dropped on to a smooth plane which is inclined at an angle \(\beta\) to the horizontal. If \(\theta_n\) denotes the angle between the plane and the direction of motion immediately after the \(n\)th bounce, show that \(\theta_n \to 0\) as \(n\to\infty\), and that \[ \theta_{n+1} = \theta_n - 2\theta_n^2 \tan\beta \] for large values of \(n\), neglecting terms of the order of \(\theta_n^3\).
A cloud of water vapour moves vertically upwards with velocity \(V\), and a spherical drop of water in the cloud moves in a vertical line under the action of gravity. The mass of the drop increases, by condensation from the cloud, at a rate \(k_1S\), where \(S\) is the surface area of the drop. Relative motion between the drop and the cloud is opposed by a force of magnitude \(k_2S\) times the relative velocity. If the drop is initially very small and moving with the cloud, show that it will begin to fall after a time \[ \frac{V}{g}\left(4+3\frac{k_2}{k_1}\right). \]