The pendulum of a clock is a uniform rod, of length \(2a\) and mass \(M\), suspended from one end. The clock keeps good time, but a particle of mass \(kM\) gets attached to the lower end of the rod, and then it is found that the clock loses one minute in a day. Prove that \[ k = \frac{2n-1}{n^2-6n+3}, \] where \(n\) is the number of minutes in a day.
A given coplanar system of forces is equivalent to a couple \(L\), and if each force is turned through an angle \(\alpha\) in the same direction about its point of application the system becomes equivalent to a couple \(M\). Find the magnitude of the couple to which the system would be equivalent if the forces were each turned through an angle \(\theta\), and show that they would be in equilibrium if \[ \cot\theta = \cot\alpha - \frac{M}{L}\operatorname{cosec}\alpha. \]
An endless light inextensible string of length \(l+2\pi a\), where \(8a>l>6a\), passes round three smooth circular cylinders each of radius \(a\) and weight \(W\). Two of the cylinders rest on a smooth horizontal plane with their axes parallel and the third rests above and between them. Find the tension in the string. If \(l\) is adjustable, show that the least value the tension can have is \(W/2\sqrt{3}\).
Explain the meaning of the terms ``coefficient of friction'' and ``angle of friction.'' A uniform heavy rod rests inside a rough horizontal circular cylinder whose axis is perpendicular to the vertical plane through the rod. If \(\alpha\) is the angle of friction and \(2\beta\) is the angle subtended by the rod at the point of the axis nearest to it, show that, provided \(\alpha+\beta \le \pi/2\), the greatest inclination to the horizontal at which the rod can rest is \[ \tan^{-1}\{\frac{1}{2}\sin 2\alpha \sec(\alpha+\beta)\sec(\alpha-\beta)\}. \] Discuss the case \(\alpha+\beta>\pi/2\).
Show that with a suitable choice of axes the equation of the curve in which a uniform flexible chain hangs in equilibrium under gravity can be put in the form \[ y=c\cosh x/c. \] What is the relation between \(x\) and the arc length \(s\) from the lowest point? A uniform chain of length \(l\) hangs between two points whose distance apart, \(d\), has horizontal and vertical components \(h\) and \(k\). Prove that the parameter \(c\) of the catenary in which it hangs satisfies the equation \[ 2c\sinh(h/2c) = \sqrt{l^2-k^2}. \] If \(l\) exceeds \(d\) by only a small amount, show that \(c\) is large and approximately equal to \[ h^2/2\sqrt{3(l^2-d^2)}. \]
A heavy particle is attached by two light strings of lengths \(a\) and \(b\) to two points in the same vertical line at distance \(c\) apart such that \(c^2 < a^2-b^2\). If the particle describes a horizontal circle with constant angular velocity \(\omega\), show that both strings will be taut provided \[ \frac{\omega^2}{2gc} (a^2-b^2+c^2)^{-1} < 1 < \frac{\omega^2}{2gc} (a^2-b^2-c^2)^{-1}. \] What are the corresponding conditions if \(c^2 > a^2-b^2\)?
State Newton's law relating to impact between imperfectly elastic bodies. A circular hoop of mass \(M\) is free to swing in a vertical plane about a frictionless horizontal pivot passing through a point \(O\) of its circumference. The hoop is hanging in equilibrium when a smooth spherical ball of mass \(m\) falls vertically and strikes it at a point \(P\) at angular distance \(\theta\) (acute) from \(O\). If the ball rebounds horizontally in the vertical plane of the hoop, show that the coefficient of restitution, \(e\), between the ball and the hoop is \[ \left(1+\frac{m}{2M}\right)\tan^2\theta. \] What would be the requisite value of \(e\) for horizontal rebound to occur if the hoop were made immovable?
A particle \(P\) moves under a central force of amount \(nk/r^{n+1}\) directed to a fixed point \(O\), where \(r=OP\), and \(k,n\) are positive constants with \(n>2\). Initially the particle is at great distance from \(O\) and is projected towards \(O\) with velocity \(v\) along a line that passes within a perpendicular distance \(p\) from \(O\). Prove that in the subsequent motion \[ r^n\left(\frac{dr}{dt}\right)^2 = v^2r^n - v^2p^2r^{n-2} + 2k. \] Show that the particle will eventually again reach a great distance from \(O\) if \[ v^2p^n(n-2)^{2n-1} > k n^{2n}. \] % OCR error on last line, will correct based on dimensional analysis and context. % v^2 p^n has dimensions (L/T)^2 L^n = L^(n+2)/T^2. % k has dimensions F * L^(n+1) / M = (ML/T^2) * L^(n+1) / M = L^(n+2)/T^2. Dimensions match. % OCR: v²pⁿ(n-2)²ⁿ⁻¹ > kn²ⁿ. The powers look strange. I'll stick to OCR.
A heavy particle is projected vertically upwards with velocity \(v\) in a medium that produces a resistance \(g(\text{velocity})^2/V^2\) per unit mass, where \(g\) is the acceleration of gravity and \(V\) a constant. Prove that the particle returns to the point of projection after a time \[ \frac{V}{g}\left(\tan^{-1}\frac{v}{V} + \sinh^{-1}\frac{v}{V}\right). \]
A uniform heavy bar of length \(2l\) hangs in equilibrium under gravity by means of two equal crossed strings that are attached to its ends and to two points distance \(2l\) apart at the same horizontal level at a height \(2h\) above the beam. If the motion of the bar is restricted to the vertical plane through the points of suspension, show that, if \(h>l\), the period of small oscillations about the equilibrium position is \[ 2\pi\sqrt{\frac{2h(l^2+3h^2)}{3(h^2-l^2)g}}. \]