Two particles \(A, B\) travel in the same sense in coplanar circular paths of radii \(a\) and \(b\) respectively with a common centre \(O\), the speeds being inversely proportional to the square root of their distances from \(O\). Prove that their relative velocity will be in the direction \(AB\) when the angle \(AOB\) is \(\cos^{-1} \sqrt{ab}/(a+b-\sqrt{ab})\). Show that whatever values \(a\) and \(b\) may have there is always a real angle \(AOB\) having this value.
Solution: Wlog, let the position of \(A\) be \((a,0)\) and the position of \(B\) be \((b \cos \theta, b \sin \theta)\), then \(\vec v_A = \binom{0}{\frac{1}{\sqrt{a}}}, \vec v_B = \binom{-\frac{1}{\sqrt{b}} \sin \theta}{\frac{1}{\sqrt{b}}\cos \theta}\), the relative velocity is therefore \begin{align*} &&\vec v_A - \vec v_B &= \binom{\frac1{\sqrt{b}}\sin \theta}{\frac{1}{\sqrt{a}} - \frac{1}{\sqrt{b}} \cos \theta} \\ &&\vec s_A - \vec s_B &= \binom{a-b \cos \theta}{-b \sin \theta} \\ \Rightarrow && \frac{1}{\sqrt{b}} \sin \theta (-b \sin \theta) &= (a-b\cos \theta)\left (\frac{1}{\sqrt{a}} - \frac{1}{\sqrt{b}}\cos \theta \right) \\ \Rightarrow && -\sqrt{b} \sin^2 \theta &= \sqrt{a} - \left ( \frac{a}{\sqrt{b}} + \frac{b}{\sqrt{a}} \right)\cos \theta + \sqrt{b} \cos^2 \theta \\ \Rightarrow && 0 &= \sqrt{a}+\sqrt{b} - \frac{a\sqrt{a}+b\sqrt{b}}{\sqrt{ab}} \cos \theta \\ \Rightarrow && \cos \theta &= \frac{\sqrt{ab}(\sqrt{a}+\sqrt{b})}{a\sqrt{a}+b\sqrt{b}} \\ &&&= \frac{\sqrt{ab}}{a-\sqrt{ab}+b} \end{align*} This is always possible, since \(a + b - \sqrt{ab} \geq \sqrt{ab} > 0 \Rightarrow 1 \geq \frac{\sqrt{ab}}{a+b-\sqrt{ab}} > 0\)
A uniform heavy flexible chain of length \(2l\) hangs over a small smooth peg and is held at rest with portions of length \(l+a\) and \(l-a\) hanging vertically on the two sides of the peg. Show that if the chain is released it will leave the peg after a time \(\sqrt{l/g} \cosh^{-1} l/a\). Find also the velocity of the chain at this instant.
A particle is projected under gravity from a point \(O\) to pass through a certain point \(P\) at distance \(R\) from it and elevation \(\alpha\) above it. Prove that its trajectory will meet \(OP\) at right angles if the speed of projection is \(\sqrt{gR(\operatorname{cosec} \alpha + 3\sin\alpha)/2}\).
A uniform circular disc of mass \(M\) is free to swing in a vertical plane about a fixed horizontal smooth pivot at a point \(C\) of its circumference. It is hanging in equilibrium when a smooth particle of mass \(m\) falls vertically and impinges on it at a point \(P\) of the circumference such that \(CP\) subtends an (acute) angle \(\theta\) at the centre of the disc. If the particle rebounds horizontally in the plane of the disc, show that the coefficient of restitution \(e\) for the impact is \(\left(1+\dfrac{2m}{3M}\right)\tan^2\theta\). For what value of \(e\) would horizontal rebound occur if the disc were made immovable?
Forces \(\lambda.OP\) and \(\mu.OQ\) act along lines \(OP\) and \(OQ\) respectively and in the directions \(OP\) and \(OQ\) (\(\lambda\) and \(\mu\) being positive constants). If \(R\) is the point of \(PQ\) such that \(\lambda.PR = \mu.RQ\), show that the resultant is along \(OR\) and of magnitude \((\lambda+\mu).OR\). A uniform heavy rod \(PQ\) of length \(l\) and weight \(W\) is suspended in equilibrium by two light strings \(OP, OQ\) of lengths \(p\) and \(q\) respectively attached to a fixed point \(O\). Show that the tensions \(T_p, T_q\) in the strings are given by \[ T_p:T_q:W = p:q:\sqrt{2p^2+2q^2-l^2}. \]
Find the position of the centre of mass of a thin uniform hemispherical shell. A hollow vessel of thin uniform material consists of a right circular cylinder of radius \(a\) and height \(h\), one end of which is closed by a plane face and the other by a hemispherical shell. Show that it will stand in stable equilibrium in a vertical position with the hemisphere in contact with a horizontal table provided \(2h < (\sqrt{5}-1)a\).
The diagram represents a plane framework of nine light rods connected at smooth pin-joints \(A, B, C, D, E\) and \(F\). The frame is symmetrical about \(AC\), which is vertical. \(AB:BC:CA=5:4:3\), and \(E\) is the mid-point of \(AB\). The framework rests on horizontal supports at \(B\) and \(D\), and vertical loads of 2000 lbs., 1000 lbs. and 2000 lbs. weight are applied at \(E, A\) and \(F\), respectively. Determine by graphical methods the stress in each of the rods. % \centerline{\includegraphics[width=0.5\textwidth]{1948_mechanics_q9.png}}
Explain what is meant by the term ``angle of friction.'' Two fixed straight wires \(OP, OQ\), each inclined at an angle \(\alpha\) to the horizontal, are on opposite sides of the vertical at \(O\) and coplanar with it, \(O\) being below \(P\) and \(Q\). A heavy uniform rod \(AB\) can slide with its ends \(A, B\) in contact with \(OP, OQ\) respectively. The angle of friction \(\lambda\) is the same at each end and \(\lambda < \alpha\). Prove that the greatest inclination to the horizontal at which the rod can rest in equilibrium is \[ \cot^{-1}[\cot 2\lambda - \cos 2\alpha \operatorname{cosec} 2\lambda]. \]
Three forces \(P, Q, R\) act along three mutually perpendicular lines \(OA, OB, OC\). Their resultant is parallel to the plane \(ABC\). Prove that \[ \frac{P}{OA} + \frac{Q}{OB} + \frac{R}{OC} = 0. \]
A fixed open cylindrical jar whose radius is \(a\) stands on a horizontal table. A smooth uniform rod of length \(2l\) (\(l>a\)) rests over the rim of the jar with one end pressing against the vertical interior surface of the jar. Prove that in the position of equilibrium the inclination \(\theta\) of the rod to the horizontal is given by the equation \[ l \cos^3\theta = 2a. \] Prove that the rod will tumble out of the jar if the inclination of the rod be less than this value of \(\theta\).