A stream of particles, of mass \(\rho\) per unit volume and moving with velocity \(v\), impinges on a fixed plane \(S\), the normal to which makes an angle \(\alpha\) with the initial velocity. The impact is frictionless and the coefficient of restitution is \(e\). Find the pressure exerted on \(S\) by the stream. Show that the loss of kinetic energy per unit volume of the impinging stream is \(\frac{1}{2}(1-e)\rho\).
Particles of a system move in one plane under forces between the particles and external forces in the plane. Prove that the rate of change of angular momentum about their centroid is equal to the resultant moment of the external forces about the centroid. A compound pendulum of radius of gyration \(k\) about the centroid \(G\) hangs from a point \(P\) at distance \(h\) from \(G\). \(P\) is forced to move along a horizontal line in the plane of the pendulum, its displacement \(x\) being a known function of time \(t\), and the inclination of \(PG\) to the downward vertical being \(\theta\). Show that $$h\cos\theta \ddot{x} + (h^2 + k^2)\ddot{\theta} + hg\sin\theta = 0.$$ Find the value of \(\ddot{x}\) that is needed to make the pendulum maintain a constant inclination \(\alpha\). Show that if \(\ddot{x}\) has this value the period of small oscillations in inclination is \(\frac{2\pi}{n}\), where \(n^2 = (hg\sec\alpha)/(h^2 + k^2)\).
An elastic string of natural length \(l\) is extended to length \(l + a\) when a certain weight hangs by it in equilibrium. This string and weight hang initially from the roof of a stationary lift. Then the lift descends, with acceleration \(f\) during time \(\tau\) and thereafter with constant speed. Prove that if \(f < \frac{1}{2}g\) the string never becomes slack. Given \(f < \frac{1}{2}g\), show that during the time \(\tau\) the amplitude of oscillation of the weight is \(af/g\) and that after the time \(\tau\) the amplitude is \(2af|\sin\frac{1}{2}n\tau|/g\), where \(n^2 = g/a\).
State the principles of conservation of linear momentum and conservation of angular momentum. Explain
Necklaces consist of \(n + 3\) beads threaded on a loop of string, without a clasp and with a negligible knot, so that the beads may move round freely. Prove that the number of distinguishable necklaces that can be made from \(n\) blue beads, 2 red ones and 1 yellow one is $$\frac{1}{2}(n + 2)^2$$ if \(n\) is even. What is the corresponding number if \(n\) is odd?
(i) Prove that $$\frac{1}{4} - \frac{1}{n+1} < \sum_{r=4}^n \frac{1}{r^2} < \frac{1}{24} - \frac{2n+1}{2n(n+1)} \quad (n \geq 4).$$ (ii) Find \(\sum_{r=1}^n \frac{1}{rx^r}\).
Solve the linear recurrence relation $$u_n = (n-1)(u_{n-1} + u_{n-2}),$$ given that \(u_1 = 0\) and \(u_2 = 1\), by writing \(u_n = v_n \cdot n!\), or otherwise. Show that as \(n \to \infty\), \(u_n/n! \to 1/e\).
Prove that $$\begin{vmatrix} (a-x)^2 & (a-y)^2 & (a-z)^2 & (a-w)^2 \\ (b-x)^2 & (b-y)^2 & (b-z)^2 & (b-w)^2 \\ (c-x)^2 & (c-y)^2 & (c-z)^2 & (c-w)^2 \\ (d-x)^2 & (d-y)^2 & (d-z)^2 & (d-w)^2 \end{vmatrix} = 0.$$
\(a\), \(b\), \(c\) are three positive numbers. Prove the inequality $$abc \geq (b + c - a)(c + a - b)(a + b - c).$$ Is the inequality $$abc \geq \kappa(b + c - a)(c + a - b)(a + b - c)$$ true for any constant \(\kappa\) greater than 1 (and all \(a\), \(b\), \(c\))?
Solution: Unless \(a,b,c\) are the sides of a triangle, the LHS is positive and the RHS is negative. Therefore wlog, \(a = x+y, b = y+z, c = z+x\) (Ravi substitution, consider the incircle). Therefore it is sufficient to prove \begin{align*} && (x+y)(y+z)(z+x) &> 2z\cdot 2x\cdot 2y = 8xyz \end{align*} but \(x+y \geq 2\sqrt{xy}, y+z \geq 2\sqrt{yz}, z+x \geq 2\sqrt{zx}\) and so taking their product we obtain the required solution. No - plugging in \(a=b=c=1\) we see that \(1 \geq \kappa\).
Four complex numbers are denoted by \(z_1\), \(z_2\), \(z_3\), \(z_4\). Show that their representative points in the complex plane are concyclic if and only if the cross-ratio $$\frac{(z_1 - z_3)(z_3 - z_4)}{(z_1 - z_4)(z_3 - z_2)}$$ is real. Use this result to show that if these points are concyclic so are the points \(1/z_2\), \(1/z_3\), \(1/z_4\).