Weights \(P\) and \(Q\) are attached to the ends of a light flexible rope which is in limiting equilibrium hanging over a rough circular cylinder, the rope lying in a plane perpendicular to the axis of the cylinder which is horizontal. If \(Q\) is on the point of ascending, what weight must be added to it so that it becomes on the point of descending?
A body consists of a uniform solid hemisphere of radius \(a\) and a uniform solid right circular cone of base radius \(a\) and height \(h\) of the same density as the hemisphere, the base of the cone coinciding with the circular face of the hemisphere. Find the greatest permissible value of \(h/a\) in order that the body may be in stable equilibrium in an upright position with the hemisphere resting on a horizontal table.
A uniform rod \(AB\) of length \(l\) lies in a horizontal position on a rough inclined plane of angle \(\alpha\) for which the coefficient of friction \(\mu > \tan \alpha\). At the end \(B\) a gradually increasing force is applied acting upwards along the line of greatest slope. If the rod starts to turn about a point \(O\) such that \(OB = p\), show that \[(p/l)^2 = (\mu - \tan \alpha)/2\mu,\] and hence show that the length of rod that begins to move upwards is less than \(l/\sqrt{2}\).
A flat plate of uniform thin material is in the form of a plane quadrilateral \(ABCD\). The diagonals meet at a point \(O\). Show that its centre of mass coincides with that of four particles each of mass \(m\) at \(A\), \(B\), \(C\), \(D\) and one of mass \(-m\) at \(O\).
A narrow straight tube of length \(2a\) has one end fixed and is made to rotate in a plane with constant angular velocity \(\omega\). A small bead is instantaneously at rest at \(t = 0\) at the mid-point of the tube, and the coefficient of friction in the tube is \(\frac{1}{3}\). If gravity can be neglected, show that the particle will reach the other end of the tube after time \((2/\omega) \log x\), where \(x\) is the larger positive root of the equation \(4x^2 - 10x^4 + 1 = 0\).
A particle of unit mass is describing an orbit, whose pedal equation is \(r = p/\sin^2 \phi\), under the influence of a central force \(F(r)\). Show that the value of the force at any point of the path is given by \[F = -\frac{1}{4} h^2 \frac{d}{dr}(r^{-2}),\] where \(r\) is the radius vector, \(p\) the perpendicular from the centre \(O\) onto the tangent, \(\phi\) is the speed of the particle, and \(h = pr\). The particle is projected from a point \(P\) under an attractive force \(2\mu r/r^3\), where \(\mu\) is a constant. Prove that if the velocity of projection has a certain value, to be found, the path will be a circle passing through the centre of force \(O\).
Two small spheres of masses \(m_1\) and \(m_2\) are in motion along the same straight line. Show that their kinetic energy may be written in the form \[\frac{1}{2}Mu^2 + \frac{1}{2}\mu v^2,\] where \(M = m_1 + m_2\), \(\mu^{-1} = m_1^{-1} + m_2^{-1}\), and \(u\) is the velocity of their centre of gravity and \(v\) is the velocity of one sphere relative to the other. If the spheres subsequently collide and their coefficient of restitution is \(e\), find the loss of kinetic energy.
The point of suspension of a simple pendulum \(AB\) of length \(l\) is \(A\), and the point \(A\) is caused to move along a horizontal straight line \(OX\) in such a way that \(OA = x(t)\) at time \(t\). If \(\theta\) is the inclination of the pendulum to the vertical, and \(g\) the acceleration of gravity, obtain the appropriate equation of motion. If \(\frac{d^2x}{dt^2}\) is constant and equal to \(f\), show that the pendulum can remain at a constant inclination \(\alpha\) to the vertical given by \(\tan \alpha = f/g\), and find the period of small oscillations about this position.
A light string \(ACE\), whose mid-point is \(C\), passes through two small smooth rings \(B\) and \(D\) at the same level and distance \(2a\) apart. At the points \(A\), \(C\), \(E\) of the string are attached masses each equal to \(m\). Initially \(C\) is at rest at \(O\) (the middle point of \(BD\), \(B\) and \(D\) are hanging vertically, and the system is set free. If the total length of the string is \(4l\), show that \(C\) will come to rest when it has fallen a distance \(4a/3\). Find also the speed of \(C\) when it has fallen a distance \(3a/4\) below \(O\).
A uniform heavy rod \(AB\) of length \(2a\) is suspended in equilibrium by two light strings \(OA\), \(OB\) each of length \(2a\). If the string \(OB\) is suddenly cut, find the initial angular acceleration of the rod and the new initial tension in the string \(OA\).