Problems

A smooth wire is bent into the form \(y=\sin x\) and placed in a vertical plane with the axis of \(x\) horizontal. A bead of mass \(m\) slides down the wire starting from rest at \(x=\frac{\pi}{2}\). Shew that the pressure on the wire as the bead passes through the origin is \(mg/\sqrt{2}\), and find the pressure as it passes through \(x=-\frac{\pi}{2}\).

A body is suspended from a fixed point by a light elastic string of natural length \(l\) whose modulus of elasticity is equal to the weight of the body and makes vertical oscillations of amplitude \(a\). Shew that, if as the body rises through its equilibrium position it picks up another body of equal weight, the amplitude of the oscillation becomes \((l^2+\frac{1}{2}a^2)^{\frac{1}{2}}\).

A machine gun of mass M contains a mass M' of bullets which it discharges at the rate \(m\) units of mass per unit time, V being the velocity of the bullets relative to the ground. Shew that, if \(\mu\) be the coefficient of friction between the gun and the ground, the whole time of recoil of the gun will be \[ (2mV - \mu g M')M' / 2\mu g m M. \]

Solve the equations:

1. [(i)] \(\tan x + \tan 2x = \tan 3x\),
2. [(ii)] \(\cos^{-1}x+\cos^{-1}(1-x)=\cos^{-1}a\).

Find the sum of \(n\) terms of the series \[ \cos\alpha + \cos(\alpha+\beta) + \cos(\alpha+2\beta) + \dots. \] Prove that the sum of the squares of the distances of a point P from the angular points of a regular polygon of \(n\) sides inscribed in a circle of radius \(a\) is \(n(a^2+c^2)\), where \(c\) is the distance of P from the centre of the circle.

In the case of a triangle with the usual notation, prove that

1. [(i)] \(r_1+r_2+r_3-r = 4R\),
2. [(ii)] \(2R\cos A = 2R+r-r_1\).

If in a triangle \(a, b, B\) are given and the conditions of ambiguity are satisfied, prove that the difference of the radii of the circles inscribed in the two triangles is \[ \tan\frac{B}{2}\left\{b^2-a^2\sin^2 B\right\}^{\frac{1}{2}}. \]

Find all the values of \[ (\cos\theta+i\sin\theta)^{\frac{1}{n}} \] where \(n\) is an integer. Find the sum to infinity of the series \[ \cos^2 x - \frac{1}{2}\sin^2 2x + \frac{1}{3}\cos^2 3x - \frac{1}{4}\sin^2 4x + \dots. \]

Prove that in general a system of co-planar forces can be reduced to single force acting at a given point together with a couple. ABCD, a uniform heavy rectangle of weight W, is freely suspended by a string attached to the angular point A. Prove that if a weight \(\frac{1}{2}W \cos 2\alpha \text{cosec}\alpha\) is attached to the angular point B, the system will rest with BD horizontal, where \(\alpha\) is the angle ADB and \(AB

State the laws of friction, and define the angle of friction. A uniform circular hoop has a weight equal to its own attached to a point of its rim and is hung over a rough horizontal peg. Prove that if the angle of friction is greater than \(\pi/6\) the system can rest with any point of the hoop in contact with the peg.

State and prove the principle of virtual work. Six equal uniform rods freely jointed at their extremities form a tetrahedron. If this tetrahedron is placed with one face on a smooth horizontal table, prove that the thrust along a horizontal rod is \(w/2\sqrt{6}\), where \(w\) is the weight of a rod.