A smooth sphere of mass \(m\) collides with another smooth sphere of mass \(m'\) at rest, and after the collision the spheres move in perpendicular directions. Assuming only (i) the conservation of momentum, (ii) that the spheres remain free from rotation, and (iii) that no kinetic energy is gained in the collision, prove that \(m'\) cannot be less than \(m\). Prove further that, if the collision is of the usual type with a coefficient of restitution \(e\), then \(em' = m\). Deduce that \(e\) cannot exceed unity.
Masses \(m_1, m_2, \dots m_n\) are attached to points of a light inextensible string which hangs in equilibrium, suspended by its two ends. If the lengths of the segments of the string are given, together with the relative positions of the two ends of the string, show how to obtain sufficient equations to determine the inclinations of the segments and the tensions in them. If the masses are each equal to \(m\) and are attached at equal horizontal intervals \(h\), show that the points of attachment lie on a parabola of latus rectum \(2hT_0/mg\), where \(T_0\) is the horizontal component of the tension. Show also that an equal parabola touches the segments of the strings at their middle points, and that the distance between the vertices of these two parabolas is \(mgh/8T_0\).
Evaluate \[ \int_{-\pi}^{\pi} \cos m\theta \cos n\theta \,d\theta \] for all non-negative integral values of \(m\) and \(n\). If the function \[ f(\theta) = \sum_{r=0}^n a_r \cos r\theta, \] where \(n\) is a positive integer and the \(a_r\) are real constants, has the property that \(f(\theta) \ge 0\) for all real \(\theta\), prove by considering the integrals \[ \int_{-\pi}^{\pi} (1 \pm \cos\theta)f(\theta)\,d\theta, \] or otherwise, that \(-2a_0 \le a_1 \le 2a_0\).
Prove that, if \(ab' - a'b \neq 0\), the locus given by \[ x = at^2 + bt + c, \quad y = a't^2+b't+c', \] where \(t\) is a parameter, is a parabola. If the line joining the points with parameters \(t_1, t_2\) is parallel to \(y=mx\), prove that \[ t_1+t_2 = (b' - mb)/(ma-a'). \] Find the equation of the locus of middle points of chords of the parabola parallel to \(y=mx\).
A small ring of mass \(m\) slides on a smooth wire in the form of the parabola \(y^2 = 4ax\), the \(x\)-axis being the downward vertical. It is connected to the focus by a light spring of natural length \(a\), and rests in equilibrium at the point \((a\lambda_0^2, 2a\lambda_0)\). Show that the modulus of the spring is \(mg\lambda_0^{-2}\). Show further that, if slightly displaced, the spring vibrates with period approximately \[ 2\pi\sqrt{\frac{a(\lambda_0^2+1)}{g}}. \]
A boy of mass \(m\) stands on the horizontal floor of a truck of mass \(M\) that is free to move on level rails. He jumps, in a vertical plane parallel to the rails, so as just to clear a vertical tail-board, which is of height \(h\) and distant \(a\) horizontally. Show that his least initial velocity \(U\), relative to the truck, is given by \(U^2 = g\{(a^2+h^2)/h + h\}\). Find also the magnitude of the least impulsive reaction between the boy and the truck, and show that its direction \(\theta\) with the horizontal is given by \(\tan 2\theta = -Ma/(m+M)h\).
By transforming to polar coordinates, or otherwise, find the area of the loop of the curve \[ x^3 + y^3 = 3axy. \]
Three fixed points \(A, B, C\) are taken on a conic. Prove that there are infinitely many triangles \(PQR\), self-conjugate with regard to the conic, such that \(P, Q, R\) lie on \(BC, CA, AB\) respectively. Prove further that \(AP, BQ, CR\) meet in a point and find the locus of this point.
Prove that, if \(G\) is the centre of gravity of a uniform plane lamina of mass \(M\), \(P\) is any point of the lamina, \(I_G\) is the moment of inertia about the line through \(G\) perpendicular to the plane of the lamina, and \(I_P\) is the moment of inertia about the parallel line through \(P\), then \[ I_P = I_G + M(PG)^2. \] A uniform heavy rod \(AB\) of length \(2a\) is freely suspended from a fixed point \(O\) by two light rods \(OA, OB\), each of length \(2a\). The system is released from rest with \(AB\) vertical. Find the greatest velocity of \(A\) in the subsequent motion.
The equations of motion of a particle of mass \(m\), moving under a force \((X,Y)\) in plane, are \[ m\frac{d^2x}{dt^2} = X, \quad m\frac{d^2y}{dt^2} = Y, \] referred to rectangular axes in the plane, with origin \(O\). Deduce that the rate of change of the moment about \(O\) of the momentum of \(m\) is equal to the moment about \(O\) of \((X,Y)\). Indicate any theorem of mechanics that is assumed in the course of the proof. Two particles of masses \(M,m\) are joined by a light inextensible string; \(M\) lies on a smooth horizontal table, and the string passes through a small hole \(O\) in the table so that \(m\) hangs below the table. Initially \(M\) is at a distance \(r_0\) from \(O\) and is moving horizontally at right angles to \(OM\) with velocity \(V_0\). Obtain an equation of the form \((dr/dt)^2 = f(r, r_0, V_0)\) for the distance \(r\) of \(M\) from \(O\) in the subsequent motion. Deduce from this equation that if \(M\) again moves at right angles to \(OM\), \(r\) must then equal one of the two values \(r_0\) and \(r_1\), where \[ r_1 = \rho\{1 + (1+2r_0/\rho)^{1/2}\} \quad \text{and} \quad \rho = MV_0^2/4mg. \] By considering the sign of \(f(r,r_0,V_0)\) show that \(r\) must lie between \(r_0\) and \(r_1\).