A stiff rod \(AB\) of length \(a\) pivots about a fixed point \(A\) and is attached by an elastic string, of unstretched length \(a\) and modulus \(E\), to a fixed point \(C\) at a distance \(a\) from \(A\). A force parallel to and in the direction of \(CA\), of magnitude equal to \(E/8a\) times the length \(BC\), is applied to the rod at \(B\), the rod being constrained to lie in a fixed plane through \(AC\). Show that there exists a position of equilibrium where the rod is not parallel to the string. Show also that this position is stable.
State the principle of virtual work. A smooth sphere of radius \(r\) and weight \(W\) rests in a horizontal circular hole of radius \(a\). A smooth string is wrapped twice round the sphere just above the hole and pulled tight. What tension in the string will just raise the sphere?
A particle \(A\) of mass \(m\), and a particle \(B\) of larger mass \(M\), are attached to the ends of a light inelastic thread which hangs over a smooth peg; the particle \(A\) is also attached to one end of a light elastic string, whose unstretched length is \(a\) and whose other end is attached to a fixed point \(C\) which is vertically below the peg. Originally the system is at rest in equilibrium, and then the stretched length of the elastic string is \((a+c)\). The system is set in motion by a downward impulse \((M+m)v\) on the particle \(B\). Show that during the subsequent motion the string and the thread both remain taut if $$v^2 < \frac{M-m}{M+m} gc.$$
A particle is projected with velocity \(v\) at an angle \(\alpha\) to the vertical from a point on a horizontal plane. The coefficient of restitution on impact is \(e < 1\), this applying to the vertical component of velocity only; the horizontal component is unaffected by impact. Find an expression for the height reached on each bounce in terms of the horizontal distance travelled by the particle. What happens to the particle for times greater than \(2v\cos\alpha/g(1-e)\)? Show also that if the coefficient of restitution is given by \(e = a/(v+a)\), where \(a\) is the vertical velocity of approach and \(a\) is a constant, the particle never stops bouncing.
Derive expressions for the radial and transverse components of acceleration of a particle in polar co-ordinates. A frictionless straight wire is being rotated in a horizontal plane about a fixed point of itself; and a bead of mass \(m\) is free to move on it. At time \(t = 0\) the angular velocity of the wire is \(\Omega\) and the bead is at rest relative to the wire, at a distance \(c\) from the centre of rotation. Show that, if the mass of the wire may be neglected, the torque which must be applied to the wire for the motion of the bead to satisfy the equation \(\ddot{r} = \text{constant}\) is $$\frac{3}{2}mc^2\Omega^2\{(\frac{1}{2}\Omega^2 t^2 + 1)\}^{\frac{1}{2}},$$ where \(r\) is the distance of the bead from the centre of rotation.
A rocket burns fuel at a rate equal to \(k\) times its instantaneous mass, the fuel being ejected with a fixed velocity \(P\) relative to the rocket. It is initially at rest on the surface of the Earth and is fired vertically upwards. The gravitational acceleration caused by the Earth may be taken to be \(ga^2/r^2\), where \(a\) is the radius of the Earth and \(r\) the distance of the rocket from the centre of the Earth. Show that, if \(kP > g\), the relation between the mass \(m\) and position \(r\) of the rocket is given by $$\log m = \log m_0 - \int_a^r \frac{k^2x}{2(x-a)(kPx-ag)} dx,$$ where \(m_0\) is the initial mass of the rocket.
The rotor shown in Fig. 1 is mounted on tapered axles which roll without slip on horizontal rails; the centre-line of the track is straight, but there are small variations in \(h\), the lateral spacing of the rails. If the rotor is rolling freely with a linear velocity \(v\) provided \(\rho = r + 3/g\), where \(r\) is the radius of the axle at the points of contact and \(k\) is the gyration of the rotor and axle about its axis. The axis of the rotor may be assumed to remain horizontal and at right angles to the centre-line of the rails throughout the motion.
Two circular flywheels, of uniform thicknesses \(h_1\) and \(h_2\), densities \(\rho_1\) and \(\rho_2\), and radii \(a_1\) and \(a_2\), are mounted on parallel shafts and are initially rotating at angular velocities \(\Omega_1\) and \(\Omega_2\) in the same sense. One of the wheels is moved so that its rim just touches the other, the axes remaining parallel. What is the ultimate change in the energy associated with the motion?
A uniform rod of length \(2a\) is supported symmetrically in a horizontal plane by two pegs distant \(2d\) apart. A second uniform rod of equal length but of mass twice that of the first is suspended by two equal inextensible vertical cords from the ends of the first rod. If one cord is cut find the smallest value of \(d\) for which the upper rod will not begin to tilt.
Find all the solutions of the equations \begin{align} x + y + z + w &= 2,\\ x^2 + y^2 + z^2 + w^2 &= 4,\\ xyzw &= -1,\\ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} + \frac{1}{w} &= 2. \end{align}