Two uniform rods \(AB, BC\) of the same material but of unequal lengths are rigidly jointed at right angles at \(B\), and rest inside a smooth sphere; the diameter of the sphere is equal to the length \(AC\). Show that in a position of equilibrium the plane of the rods is vertical, and find the inclinations of the rods to the vertical.
A rod \(AB\) of length \(L\) is suspended from two points on the same horizontal level by two vertical strings of natural length \(l\) and modulus of elasticity \(\lambda\) attached to the ends. If the line density of the rod increases uniformly from \(\rho\) at \(A\) to \(\rho'\) at \(B\), show that the rod is inclined to the horizontal at an angle \(\sin^{-1}\left\{\frac{lg(\rho'-\rho)}{6\lambda}\right\}\).
Prove the formulae \(s=c\tan\psi\), \(y=c\sec\psi\) for a catenary. A heavy string has one end attached to a small heavy body near the edge of a rough horizontal table, the coefficient of friction being \(\mu\). The string passes over a smooth peg which is fixed at the same horizontal level as the table, and then hangs vertically. If the body is on the point of motion, show that the angle \(\alpha\) which the string makes with the horizontal at the body or the peg is given by \[ \frac{\cos\alpha - \mu\sin\alpha}{1+2\sin\alpha} = \mu\sigma, \] where \(\sigma\) is the ratio of the weight of the body to the weight of the string.
Five equal uniform rods of weight \(w\) freely jointed together to form a convex pentagon hang from one of the vertices, and a string connects this point with the middle point of the opposite side. Show that the tension in the string is \[ \frac{2w(2\tan\alpha+\tan\beta)}{\tan\alpha+\tan\beta}, \] where \(\alpha, \beta\) are the angles which the upper and lower rods respectively make with the vertical.
(a) Find the centre of gravity of the portion of a uniform spherical shell contained between two parallel planes. (b) Find the centre of gravity of a triangle formed by three uniform rods of the same material.
A wedge of angle \(\alpha\) whose upper face is a rectangle \(ABCD\) and base a rectangle \(ABEF\) moves on a horizontal table with velocity \(v\) in a direction parallel to \(BE\). A particle moves on the upper face along the line \(AC\) with velocity \(V\) relative to the wedge. Denoting the angle \(CAB\) by \(\beta\), find the magnitude of the velocity of the particle relative to an observer ascending vertically with velocity \(W\). Show also that in the case \(W=v\cot\alpha\), the answer is the same as if the particle were moving, still with velocity \(V\), along the line \(AB\) on the wedge.
Explain what is meant by simple harmonic motion. A smooth light pulley is suspended from a fixed point by a spring of natural length \(l\) and modulus of elasticity \(\lambda\). If masses \(m_1\) and \(m_2\) hang over the pulley, show that the pulley executes a simple harmonic motion about a point whose depth below the point of suspension is \(l\left\{1+\frac{4m_1m_2}{m_1+m_2}\frac{g}{\lambda}\right\}\).
A particle whose mass at time \(t\) is \(m_0(1+\alpha t)\) is projected vertically upwards at time \(t=0\), with velocity \(v\), the added mass being picked up from rest. Show that it rises to a height \[ \frac{g+2\alpha v}{4\alpha^2}\log\left(1+\frac{2v\alpha}{g}\right) - \frac{v}{2\alpha}. \]
Find the radial and transverse accelerations of a particle moving in a plane, referred to polar coordinates. A particle of unit mass is attached to one end of an elastic string of natural length \(l\) and modulus of elasticity \(\lambda\) whose other end is fixed. The particle is held so that the string is just tight, and projected at right angles to the string with velocity \(2\sqrt{\frac{\lambda l}{3}}\). Show that the greatest length of the string in the resulting motion is \(2l\).
Define the moment of inertia of a rigid body about an axis, and find the kinetic energy when the body rotates with angular velocity \(\Omega\) about that axis. A uniform circular disc of mass \(M\) and radius \(a\) is free to rotate in a vertical plane about a point \(P\) on its rim. A particle of mass \(2M\) is fastened at a point \(Q\) on the diameter through \(P\), and the system makes small oscillations about the position of stable equilibrium. Find the position of \(Q\) in order that the length of the simple equivalent pendulum may be a minimum.