Two uniform smooth spheres of equal mass experience an elastic collision (coefficient of restitution equal to unity). Initially one of the spheres is at rest but free to move in any direction. Show that after the collision the directions of motion of the two spheres are at right angles to each other. Discuss the special case of a head-on elastic collision. Show further that, if the mass of the sphere initially in motion is the less in the ratio \(1-\epsilon : 1\) (\(\epsilon\) small), and if \(\theta\) is the deflection in its motion, then, to the first order in \(\epsilon\), the angle between the directions of motion after collision exceeds a right angle by \[ \frac{1}{2}\epsilon\tan\theta. \]
A particle is projected from a point \(P\) in an attractive field of force \(\mu/r^5\), where \(r\) is the distance from the fixed centre of attraction \(O\). Show that when the velocity of projection is \((\mu)^{1/2}/OP^2\) the orbit is a circle passing through \(O\).
A uniform rod \(AB\) of mass \(2M\) and length \(2a\) is smoothly hinged at its end \(B\) to a point on the rim of a uniform circular disk of mass \(M\) and radius \(r\). The rod and disk are laid on a smooth horizontal table so that the direction of \(AB\) passes through the centre of the disk. A horizontal impulse \(P\) is applied at \(A\) at right angles to \(AB\). Show that the kinetic energy produced is \(\frac{9}{10}\frac{P^2}{M}\) and that the impulsive reaction at the hinge is \(P/5\).
A uniform circular ring whose centre is \(O\) is rotating in its own plane with angular velocity \(\omega\) about a fixed point \(A\) on the ring. The point \(A\) is suddenly released and a second point \(B\) on the ring fixed, where \(\angle AOB = \theta\). Find the new angular velocity about \(B\).
A uniform thin rod of length \(2a\) is supported by two small rough pegs at different levels. The upper peg lies above and the lower peg below the rod. The pegs are at a distance \(c( < a)\) apart, and the line joining them makes an angle \(\alpha\) with the horizontal. The coefficient of friction at the upper peg is \(\mu_1\) and at the lower peg \(\mu_2\). Find the greatest value of \(\alpha\) at which equilibrium can be maintained.
Six equal uniform bars, each of weight \(W\), are freely jointed together so as to form a regular hexagon \(ABCDEF\), which hangs from the point \(A\) and is kept in shape by strings \(AC, AD, AE\). Find the tensions in these strings.
Define the ``bending moment'' at a point of a beam, and explain its physical meaning. A curved rod of length \(l\) in the form of an arc of a circle of radius \(R\) has its ends connected by a string in tension. Prove that the bending moment at any point is proportional to \(\sin\frac{\alpha}{2}\sin\frac{\beta}{2}\), where \(\alpha\) and \(\beta\) are the angular distances of the point in question from the ends of the rod. Given that the rod breaks when the bending moment exceeds the value \(M_0\), find the tension in the string when the rod is about to break.
Examine the stability of a plank of thickness \(2a\) which rests horizontally across the top of a fixed horizontal rough cylinder of radius \(a\). What is the effect of giving the plank a slight bow so that it is concave towards the cylinder?
Show that the tensions at two points of a coplanar light string wrapped around a rough cylinder are related by \[ T_2=T_1 e^{\mu\theta} \] when the string is about to slip, where \(\mu\) is the coefficient of friction and \(\theta\) is the angle between the tangents at the two points. Two weights \(P, Q\) hang in limiting equilibrium from a light string which passes over a rough circular cylinder in a plane perpendicular to the axis. If \(P\) be on the point of descending, what is the maximum weight that may be added to \(Q\) without causing it to descend?
A particle \(P\) is projected from a point \(O\) with velocity \(V\). Show that, when the line \(OP\) makes an angle \(\phi\) with the upward vertical through \(O\), the distance \(OP\) cannot exceed \(V^2/\{g(1+\cos\phi)\}\). A gun, of constant muzzle velocity, is sited at a point \(O\) of a plane hillside, which makes an angle \(\alpha\) with the horizontal. The gun can fire in any direction and at any elevation; show that the region of the hillside within range has the shape of an ellipse with focus at \(O\) and eccentricity \(\sin\alpha\).