Determine the asymptotes of the curve \[ (y-1)^2(y^2-4x^2) = 3xy. \] Investigate on which sides of the asymptotes the corresponding branches of the curve lie and trace the curve.
If \(u = \int_0^\theta \frac{d\theta}{\cos\theta}\), show that \(\theta = \int_0^u \frac{du}{\cosh u}\), and if \[ \int_0^\theta \frac{d\theta}{\cos\theta} + \int_0^\phi \frac{d\phi}{\cos\phi} = \int_0^\psi \frac{d\psi}{\cos\psi}, \] show that \[ \cos\psi = \frac{\cos\theta\cos\phi}{1+\sin\theta\sin\phi}. \]
The centres of two spheres of masses \(m_1, m_2\) are moving in the same straight line so that the first overtakes the second. State the reason for the assertion that the total momentum is unaltered by the impact, and shew that in general Kinetic Energy is lost. Criticise the argument that since `action and reaction are equal and opposite' and the points of application of the action and reaction move the same distance while the spheres are in contact therefore the action and reaction do equal and opposite amounts of work and therefore there is no change in the Kinetic Energy.
A light inextensible string \(BC\) joins the ends of two uniform rods \(AB\) and \(CD\) which are of the same length. The system is placed on a smooth horizontal table so that \(AB, BC\) and \(CD\) form three sides of a rectangle, and an impulse \(I\) in the direction \(AD\) is then applied to the rod \(AB\) at \(A\). Prove that the initial velocity of \(D\) is \(I/2m\), \(m\) being the mass of each rod.
A curve is traced by a point on the circumference of a circle radius \(a\) which rolls on the outside of a fixed circle of radius \(3a\). Shew that the area of the curve so traced is \(20\pi a^2\), and find the whole length of the curve.
\(AB\) and \(CD\) are perpendicular diameters of a circle. Find the mean value of the distance of \(A\) from points on the semicircle \(CBD\) and also the mean value of the reciprocal of that distance. Shew that the product of these means is \[ \frac{8\sqrt{2}\log_e(1+\sqrt{2})}{\pi^2}. \]
A light rod \(AB\) is suspended from a point \(O\) by strings \(OA\) and \(BO\) of lengths \(a_1\) and \(a_2\) respectively. A particle of weight \(w_1\) is attached to the point \(A\) and another particle of weight \(w_2\) is attached to the point \(B\). If \(\theta_1\) and \(\theta_2\) are the inclinations of the strings \(OA\) and \(BO\) to the vertical, shew that \[ a_1 w_1 \sin\theta_1 = a_2 w_2 \sin\theta_2, \] and that \[ \frac{T_1}{a_1 w_1} = \frac{T_2}{a_2 w_2}, \] where \(T_1\) and \(T_2\) are the tensions in the strings \(OA\) and \(BO\) respectively.
A heavy bar of length \(a\) rests inclined at an angle \(\theta\) to the vertical with the lower end on rough horizontal ground, the coefficient of friction between the bar and the ground being \(\mu\). A man \(A\) pushes horizontally at the upper end, and another man \(B\) presses vertically down on the bar at a point \(P\) distant \(a\mu\cot\theta\) from the lower end. Shew that if the centre of gravity \(G\) lies below \(P\), \(B\) can maintain equilibrium no matter how hard \(A\) pushes; but that if \(G\) lies above \(P\), \(B\) cannot maintain equilibrium.
A heavy beam \(AB\) of length \(l\) and weight \(w\) is freely hinged at \(A\); it is supported by resting on the end of a light inclined rod of length \(c\) freely hinged at a point \(C\) in the same vertical plane as the beam. \(C\) is on the same horizontal level as \(A\) and the distance \(CA=d\). Also \(l>d+c>2c\). The angle of friction between the beam and the rod is \(\phi\). Shew that there are two positions of equilibrium when the friction is limiting, and that the work done against friction in applying a couple to the rod and slowly moving it from one of these positions to the other is \[ \frac{2acw \sin^2\phi}{d \cos\phi}, \] where \(a\) is the distance of the centre of gravity of the beam from \(A\). Assume that during the motion the friction is limiting.
Two uniform rods \(AO\) and \(OB\) each of length \(2l\) and weight \(w\) are freely jointed together at \(O\). Their ends \(A\) and \(B\) are free to move without friction along a horizontal groove. A third rod \(CO\) of length \(l'\) and weight \(w\) lies in a vertical plane perpendicular to the groove and has its end \(C\) freely hinged to a point in the vertical plane containing the groove and at a distance \(d\) from it. A weight \(w\) is suspended from \(O\). The middle points of the rods \(AO\) and \(OB\) are connected by an inelastic string of length \(x\). Shew that the tension in the string is equal to \(\frac{5}{2}\frac{xw}{d}\).