A bead of weight \(w\) is threaded on a smooth circular wire of radius \(a\) which is fixed in a vertical plane. A light elastic string of unstretched length \(a\) and modulus of elasticity \(\lambda\) joins the bead to the highest point of the wire. If \(\lambda > 2w\), prove that there are three positions of equilibrium of the bead on the wire (with the string stretched), and discuss the stability of each position.
A particle is projected with velocity \(V\) from a point \(P\) so as to pass through a small ring at a horizontal distance \(a\) and a vertical distance \(b\) (upwards) from \(P\). Prove that the angle of projection \(\theta\) must satisfy \[ \tan^2\theta - \frac{2V^2}{ga}\tan\theta + \left(\frac{2V^2b}{ga^2}+1\right)=0. \] Hence find the least possible value of \(V\) and the corresponding angle of projection, and prove that for these conditions of projection the range on the horizontal plane through \(P\) is \[ a\left(1+\frac{b}{\sqrt{(b^2+a^2)}}\right). \]
A spherical raindrop of initial radius \(a\) falls from rest under gravity. Its radius increases with time at a constant rate \(\mu\) owing to condensation from a surrounding cloud which is at rest. Find the distance fallen by the raindrop after time \(t\). How would the equation of motion of the raindrop be affected if the cloud was falling with vertical velocity \(V\)?
Prove that the moment of inertia of a uniform spherical shell of radius \(a\) and mass \(M\) about a diameter is \(\frac{2}{3}Ma^2\). A uniform spherical shell of radius \(a\) and mass \(M\) is released from rest on a rough plane which is inclined at an angle \(\alpha\) to the horizontal. Prove that the shell will roll without slipping provided that the coefficient of friction is greater than \(\frac{2}{5}\tan\alpha\).
A uniform circular disc of radius \(r\) and mass \(M\) rests with one face in contact with a smooth horizontal table. An impulse \(J\) in the plane of the table is applied to a point \(P\) on the circumference of the disc in a direction which makes an angle \(\theta\) with the diameter through \(P\). Find the resulting velocity of \(P\) and the kinetic energy of the disc.
Two fixed points \(A\) and \(B\) are on the same horizontal level and a distance \(2l\) apart. They are joined by an elastic string, of natural length \(2l\) and modulus of elasticity \(\lambda\), which carries a particle of mass \(m\) at its mid-point. The particle is released from a point vertically below the mid-point of \(AB\). Prove that its equation of motion is \[ m\ddot{y} = mg - \frac{2\lambda y}{l}\left(1-\frac{l}{\sqrt{(l^2+y^2)}}\right), \] where \(y\) is its distance at time \(t\) below the mid-point \(AB\). Given that \(\lambda=mg/\sqrt{3}\), prove that the particle can rest in equilibrium when \(y=l\sqrt{3}\), and find the period of small vertical oscillations about the position of equilibrium.
Explain how the resultant of a three-dimensional system of forces may in some circumstances be a couple, and show that in this case the straight lines about which the system has zero moment must be parallel to the same plane. Prove that forces acting along and inversely proportional to the lengths of the perpendiculars from the vertices of any tetrahedron on to the opposite faces are in equilibrium.
A thin uniform rigid rod of weight \(W\) resting on a rough peg at \(A\) and supported from above by a similar peg at \(B\) is in equilibrium under its own weight and the reactions of the pegs. If the coefficient of friction is the same at both pegs, show that it cannot be less than \(\mu_0 = \frac{AB \tan\theta}{AC+BC}\), where \(C\) is the centre of the rod and \(\theta\) its inclination to the horizontal. An additional weight \(W'\) is attached to a certain point \(D\) of the rod such that equilibrium is still maintained but with increase of \(W'\) is eventually destroyed. Show that the rod will tend to turn or slip as the coefficient of friction is greater than or less than \(\tan\theta\).
A heavy uniform horizontal beam of length \(2l\) rests symmetrically on two supports which are at a distance \(2a\) apart on a horizontal level. Illustrate, by a simple sketch, the value of the bending moment for every point of the beam. If \(a=(4-2\sqrt{3})l\), find the points on the beam where the bending moment is zero.
A uniform heavy flexible chain hangs under gravity with its ends attached to light smooth rings which can slide on a horizontal rod. The rod is rotated about a vertical axis intersecting it, with constant angular velocity \(\omega\), and the chain is at relative rest in a vertical plane with the axis of rotation as an axis of symmetry. \(x\) is measured horizontally from the axis of rotation and \(y\) is measured vertically upwards from the lowest point of the chain. Prove that the tension at any point of the chain is given by \[ T=T_0 + pgy - \frac{1}{2}p\omega^2x^2, \] where \(T_0\) is the tension at the lowest point and \(\rho\) is the line density of the chain. Prove that the total length of the chain is \[ 2y_1 - \frac{\omega^2}{g}x_1^2 + \frac{2T_0}{\rho g}, \] where \(x_1, y_1\) are the values of \(x\) and \(y\) at one of the rings.