\(n-1\) particles are attached to a light inextensible string \(A_0 A_n\) at points \(A_1, A_2, \dots, A_{n-1}\), where \(A_0A_1 = A_1A_2 = \dots = A_{n-1}A_n\). A small ring of weight \(w_n\) attached to the string at \(A_n\) can move without friction on a fixed vertical wire \(OA_n\). The system is in equilibrium with the end \(A_0\) fixed at the same horizontal level as \(O\) and \[ OA_0 = OA_1 = OA_2 = \dots = OA_n. \] Obtain in terms of \(r\) and \(n\) the ratio of the weight of the particle at \(A_r\) to the total weight. By considering the position of the centre of gravity, show that \[ \sum_{r=1}^n \cos \frac{r\pi}{2n} \operatorname{cosec} (2r+1)\frac{\pi}{4n} \operatorname{cosec}(2r-1)\frac{\pi}{4n} = \operatorname{cosec}\frac{\pi}{4n} \left(\operatorname{cosec}\frac{\pi}{4n} - \sec\frac{\pi}{4n}\right). \]
Prove that, in two dimensions, a system of forces is in general equivalent to a force acting in a given direction together with a force acting in a given line not in the given direction. What are the exceptional cases which may arise? A, B, C, D, E, F, G, H are the vertices, taken in order, of a regular octagon. Forces of magnitudes 1, 2, 2 lb. wt. act in AB, BD, DH respectively. Show that these forces are equivalent to a force in FE together with a force of magnitude \(2\sqrt{2}-\sqrt{2}\) lb. wt. parallel to AH, and find the magnitude of the force in FE.
The ends of a light elastic string of natural length \(2a\) and modulus of elasticity \(\lambda\) are attached to a uniform rod BC of length \(2a\) and weight \(W\) at the end B and at a point D. The rod is suspended in equilibrium with the string passing over a smooth peg A. If the distances of the mid-point of the rod from A and D are equal, prove that \(\frac{W}{\lambda} < 2-\sqrt{2}\) and that the distance BD exceeds \(2^{\frac{3}{4}}a\).
A uniform cube of edge \(a\) and weight \(w\) rests on a rough horizontal plane. A uniform rod of length \(2a\) and weight \(W\) rests in limiting equilibrium against the cube the vertical plane through the rod bisecting four of the edges of the cube. The rod is inclined to the horizontal at an angle \(\tan^{-1}\frac{4}{3}\), and the coefficient of friction between the rod and the cube is \(\frac{3}{2}\). Find the coefficient of friction between the rod and the horizontal plane. Show further that \(w > \frac{2}{5}W\) and that the coefficient of friction between the cube and the horizontal plane exceeds \(\frac{15W}{40w+12W}\).
At time \(t\) a particle moving in a straight line has speed \(v\) and its distance from its position when \(t=0\) is \(s\). If \(vt-3s\) is proportional to \(u-v\), where \(u\) is constant, find the acceleration in terms of \(v\). Show that, if the acceleration is \(a\) when \(t=0\), then the acceleration is \(2a\) when \(s = \frac{14u^2}{3a}\).
A smooth wire is bent into the form of a circle of radius \(a\) and is held with its plane inclined to the horizontal at an angle \(\alpha\). A small bead, projected with speed \((\frac{8}{3}ga \sin\alpha)^{\frac{1}{2}}\) from the lowest point of the wire, moves on the wire. Prove that if \(\alpha > \frac{\pi}{6}\) the resultant reaction between the bead and the wire is horizontal in two and only two positions of the bend.
When unstretched, a light elastic string is of length \(2a\) and has a particle attached to it at its mid-point. When the ends of the string are fixed at two points a distance \(4a\) apart in the same vertical line and the particle is released from rest at the lower end it rises through a height \(3a\) before coming instantaneously to rest again. Show that the time taken to rise this height is \(\left(\frac{5a}{12g}\right)^{\frac{1}{2}}(\pi - \cos^{-1}\frac{1}{7} + 2\cos^{-1}\frac{1}{13})\).
A particle is projected from a given point O at an elevation \(\alpha\) and moves freely under gravity. If \(h\) is the greatest height above O attained, show that when the direction of motion has been turned through an angle \(\beta (<\frac{\pi}{2}+\alpha)\) the height of the particle above O is \(h(1-\cot^2\alpha \tan^2(\alpha-\beta))\) and the elevation of the particle from O is \(\tan^{-1}\frac{1}{2}(\tan\alpha + \tan(\alpha-\beta))\).
A particle moves inside a fine smooth straight tube which is made to rotate about a point O of itself with constant angular velocity in a horizontal plane. Initially the particle is at relative rest at a distance \(b\) from O and it subsequently rebounds from a closed end of the tube at a distance \(a\) from O. If \(e\) is the coefficient of restitution, show that after \(n\) rebounds the least distance of the particle from O is \(\{a^2(1-e^{2n}) + b^2e^{2n}\}^{\frac{1}{2}}\).
Find the moment of inertia of a uniform thin spherical shell of mass \(m\) and radius \(a\) about a diameter. If the shell is smoothly pivoted at a point of itself and if the pivot cannot support a load greater than \(\frac{1}{2}mg\), show that the greatest horizontal impulse which can be applied to the shell along a diameter when the shell is hanging in equilibrium is \(m(5ga)^{\frac{1}{2}}\) if the shell is not to break away from the pivot. It may be assumed that the pivot can sustain the initial impulsive reaction.