Problems

Solve the equations:

1. [(i)] \(\tan^{-1} \dfrac{1-x}{1+x} = \frac{1}{2}\tan^{-1} x\);
2. [(ii)] \(4\sin 3\theta - 3\sin 4\theta = 0\).

Prove that (i) if \(\theta+\phi+\psi = \pi/2\), \[ \sin^2\theta + \sin^2\phi + \sin^2\psi + 2\sin\theta\sin\phi\sin\psi = 1; \] (ii) if \(\theta+\phi+\psi = \pi/4\), \[ \frac{(1-\tan\theta)(1-\tan\phi)(1-\tan\psi)}{(1+\tan\theta)(1+\tan\phi)(1+\tan\psi)} = \frac{\sin 2\theta + \sin 2\phi + \sin 2\psi - 1}{\cos 2\theta + \cos 2\phi + \cos 2\psi}. \]

In a triangle prove that

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

If the distance between \(A\) and the orthocentre of a triangle \(ABC\) is equal to \(r\), prove that the circumscribed circle cuts orthogonally the escribed circle opposite \(A\).

Find the sum to infinity of the series \(1+2x\cos\theta + 2x^2\cos 2\theta + 2x^3 \cos 3\theta + \dots\) and deduce the series for \(2\cos n\theta\) in descending powers of \(2\cos\theta\).

Forces represented by \(\lambda OA, \mu OB\) act at \(O\) towards \(A\) and \(B\) respectively; prove that their resultant is represented by \((\lambda+\mu)OG\), where \(G\) is the centroid of particles of masses proportional to \(\lambda\) at \(A\) and \(\mu\) at \(B\). Particles of masses \(m_1, m_2, m_3\) are placed at the vertices \(A,B,C\) of a triangle and attract each other with forces proportional to their masses and inversely proportional to the square of the distances between them. Prove that the resultant forces on the particles act towards a point \(G\), which is the centroid of masses \(m_2 a^3, m_3 b^3, m_1 c^3\) at \(A,B,C\) respectively, and that their magnitudes are in the ratios \(m_1 a^3 AG : m_2 b^3 BG : m_3 c^3 CG\).

State the laws of statical friction. At points \(A, A', A''\) on a rough horizontal plane are placed weights \(w, w', w''\), (\(w

Define angular velocity, and explain how to find the angular velocity of the line joining two points whose motions are given. A particle \(P\) is describing a parabola freely under gravity. Shew that the angular velocity of the line joining \(P\) to the focus is \(2gu/v^2\), where \(v\) is the velocity of the particle at \(P\) and \(u\) is the horizontal component of \(v\).

An engine weighing 96 tons, of which 40 tons are carried by the driving wheels, exerting a uniform pull gives a train a velocity of 25 miles per hour after travelling for 50 seconds from rest against a resistance of 10.5 lb. weight per ton. If the friction between the driving wheels and the rails is 0.2 times the pressure, find the tension in the coupling between the engine and the first carriage.

Two spherical particles moving in a given manner impinge, write down equations to determine the motion after impact. Two equal smooth spheres \(A, B\) lie in contact on a smooth horizontal plane; a third equal sphere \(C\) is projected with a given velocity along the table so as to strike \(A\) and \(B\) simultaneously. Find the velocities of each sphere after impact and shew that the sphere \(C\) passes through and beyond the two spheres \(A\) and \(B\) if the coefficient of restitution between the spheres is \(<1/9\).

State the principles of the conservation of energy and of angular momentum. A light string passing through a smooth ring at \(O\) on a smooth horizontal table has particles each of mass \(m\) attached to its ends \(A\) and \(B\). Initially the particles lie on the table with the portions of string \(OA, OB\) straight and \(OA=OB\). An impulse \(P\) is applied to the particle \(A\) in a direction making \(60^\circ\) with \(OA\). Prove that when \(B\) reaches \(O\) its velocity is \(P\sqrt{22}/8m\).