A heavy flywheel consists of a uniform circular disc of radius \(a\) and mass \(M\) which can rotate about an axle through the centre perpendicular to its plane. This axle can be engaged with a coaxial light cylindrical drum of radius \(b\), around which is wound one end of a light rope. The rope supports a cage of mass \(m\) which hangs freely in a vertical shaft. When the flywheel is running freely with angular velocity \(\omega\) it is suddenly engaged with the drum so that the cage begins to rise. Find the height through which the cage rises (assuming that the rope is long enough to prevent the cage hitting the drum).
Prove that a given force acting in the plane of a triangle is equivalent to three forces acting along the sides of the triangle. Find the magnitudes of the three forces if the lengths of the sides of the triangle are 3, 4, 5, while the given force is of magnitude \(F\) and acts in the line bisecting the side of length 4 at right angles.
A tripod formed of three uniform rods \(OA, OB, OC\), which are of the same weight and of the same length \(2a\) and are freely jointed together at \(O\), rests on a rough horizontal plane so that the feet \(A, B, C\) form an equilateral triangle. If the coefficient of friction is \(\frac{1}{4}\), prove that the least possible height of \(O\) above the plane is approximately \(1.79a\).
A wedge is cut from a uniform solid circular cylinder of radius \(a\) by two planes inclined at an angle \(\alpha\). One plane is perpendicular to the axis of the cylinder, and the line of intersection of the planes touches the surface of the cylinder. Prove that the mass-centre of the wedge is at a perpendicular distance \(\frac{3}{8}a \tan\alpha\) from the circular base of the wedge, and explain why the mass-centre is not near the centre of the base when \(\alpha\) is small.
A uniform bar \(AB\) of length \(l\) and weight \(w\) per unit length is attached to a fixed smooth hinge at \(A\) and is kept horizontal by a light chain of length \(2l\) which joins \(B\) to the point vertically above, and distant \(\sqrt{3}l\) from, \(A\). Find the thrust, shearing force and bending moment at any point of the bar.
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.