Problems

A body of mass \(M\) moves in a straight line under the action of a force which works at constant power \(P\), and against a resistance equal to \(P/C^2\) times the velocity, \(C\) being constant. If the body starts from rest, shew that after time \(t\) its velocity \(u\) is given by \[ u^2 = C^2 [1 - e^{-2Pt/(MC^2)}] \] and that the distance \(s\) traversed is equal to \[ \frac{MC^3}{2P} \log \frac{C+u}{C-u} - \frac{MC^2}{P} u. \]

A bead of mass \(m\) slides on a smooth straight wire inclined at an angle \(\alpha\) to the vertical, and is attached to one end of a long light inextensible string, which passes over a fixed smooth peg vertically above a point \(A\) of the wire at a height \(h\) and carries a mass \(M\) hanging freely at its other end. The vertical portion of the string passes the wire without interference. Shew that, if the bead is released from rest at \(A\), it will move upwards if \(M>m\) and downwards if \(M

State the principle of the conservation of linear momentum for the motion of any number of particles. \par Four particles of equal mass at four consecutive vertices \(A, B, C, D\) of a regular hexagon are on a smooth horizontal table and are connected by light inextensible strings \(AB, BC, CD\), which are taut. The particle at \(A\) is projected with velocity \(V\) parallel to and in the same sense as \(DC\). Find the velocity of the particle at \(D\) just after the string \(AB\) becomes taut again.

A particle moves in a straight line through a fixed point \(O\) so that, if \(x\) is its distance from \(O\) at time \(t\), its equation of motion is \[ \frac{d^2x}{dt^2} + n^2x = n^2l. \] Shew that its motion is a simple harmonic oscillation about the point \(x=l\). The equation of motion for a simple harmonic oscillation about \(O\) may be assumed to be \[ \frac{d^2x}{dt^2} + n^2x = 0. \] A spring of natural length 1 foot, negligible inertia and modulus 10 lb. weight has one end \(S\) fixed, and a mass of 2 lb. is fixed to the other end. The spring is parallel to a rough horizontal table on which the mass can move in a straight line through the fixed end of the spring, the coefficient of friction between the mass and the table being \(\frac{1}{4}\). If \(A\) and \(B\) are the positions of limiting equilibrium, \(SA\) being greater than \(SB\), find the lengths of \(SA\) and \(SB\). \par The spring is extended to a length 1.75 feet and the mass is released from rest. Shew that the subsequent motion of the mass is a simple harmonic half-oscillation about \(A\), followed by a half-oscillation about \(B\), and so on, and determine (i) the length of the spring when the mass finally comes to rest, and (ii) the total number of half-oscillations performed.

A particle of mass \(m\) is attached to one end of a light string, the other end of which is fastened to a ring of mass \(m\) which slides on a fixed rough horizontal rod. The system is released from rest with the string taut and along the rod. Shew that in order that the ring should not slide on the rod during the ensuing motion the coefficient of friction between the ring and the rod must be not less than \(\frac{3}{4}\).

The cubic polynomial \(f(x) = x^3+bx+1\) has the roots \(\alpha, \beta, \gamma\). Find, in terms of \(b\), the coefficients of the polynomial \[ g(x) = \{(\beta+\gamma)x - \beta\gamma\}\{(\gamma+\alpha)x - \gamma\alpha\}\{(\alpha+\beta)x - \alpha\beta\}. \]

The quartic equation \[ x^4 + ax^3 + bx^2 + cx + d = 0 \] has four real roots. Prove that

1. [(i)] \(8b < 3a^2\);
2. [(ii)] \(8bd < 3c^2\);
3. [(iii)] \(8bd=3c^2\), if \(8b=3a^2\);
4. [(iv)] \(c=d=0\), if \(8b<3a^2\), \(8bd=3c^2\).

(i) If \(\alpha+\beta+\gamma = \frac{1}{2}\pi\), prove that \[ (\sin\alpha+\cos\alpha)(\sin\beta+\cos\beta)(\sin\gamma+\cos\gamma) = 2(\sin\alpha\sin\beta\sin\gamma + \cos\alpha\cos\beta\cos\gamma). \] (ii) If \(\alpha+\beta+\gamma=\pi\), \(\lambda+\mu+\nu=0\), prove that \[ \frac{\lambda}{\cos^2\alpha} + \frac{\mu}{\cos^2\beta} + \frac{\nu}{\cos^2\gamma} = 2 \tan\alpha\tan\beta\tan\gamma \left( \frac{\lambda}{\sin 2\alpha} + \frac{\mu}{\sin 2\beta} + \frac{\nu}{\sin 2\gamma} \right). \]

Evaluate \[ S_N = \sum_{\nu=1}^N e^{\frac{\nu x}{N}} \cos \frac{\nu y}{N}. \] Find the limit, as \(N\) tends to infinity, of \(\frac{1}{N}S_N\), and compare its value with \(\int_0^1 e^{tx} \cos(ty) dt\).

Find all maxima and all minima of the two functions \[ y = e^{-\sqrt{3}x} \sin^3 x \] and \[ y = \int_x^\infty \frac{\sin\xi}{\xi(1+\xi^2)} d\xi. \]