Twelve identical uniform rods, each of weight \(w\), are freely jointed to form a regular octahedron (a figure with eight equal faces, each an equilateral triangle). The octahedron is suspended from one vertex and a weight \(W\) is hung from the opposite vertex. Find the thrust in each horizontal rod.
Calculate the position of the centroid of a uniform hemisphere. A solid is shaped by cutting out from a uniform hemisphere of radius \(R\) a sphere of radius \(\frac{1}{2}R\). This solid rests with its plane face in contact with a rough inclined plane, and a gradually increasing force is applied at the pole of the hemisphere in the direction parallel to the line of greatest (upward) slope of the plane. The coefficient of friction is \(\mu\) and the angle of inclination of the plane is \(\alpha\). Show that the solid slides or tilts first according as $$\mu \lessgtr 1 - \frac{3}{2}\tan\alpha.$$
A coplanar system of forces acts on a rigid body. Show that the system is equivalent either to a single force or to a single couple. (State clearly the assumptions which are needed in the course of the proof.) Show also that the work done by the system of forces in an arbitrary infinitesimal displacement of the rigid body is equal to the work done by the equivalent force or couple.
Prove the parallelogram law of addition of velocities. An aeroplane flies on a level course at constant speed \(u\) in still air. When it flies in a wind of speed \(v\) (<\(u\)), the direction of its motion relative to the moving air makes an angle \(\theta\) with the direction of the wind; its resultant direction relative to the ground makes an angle \(\theta - \alpha\) with the direction of the wind. Derive a formula for \(\alpha\) in terms of \(u\), \(v\) and \(\theta\). Show that when \(\theta\) is such that \(\alpha\) takes its maximum value, the speed of the aeroplane relative to the ground is the geometric mean of its greatest and least speeds relative to the ground.
A particle \(P\) of unit mass moves on a smooth horizontal plane on which \(Ox\), \(Oy\) are fixed rectangular cartesian axes. \(P\) is attracted towards \(O\) with force \(n^2r\), where \(r\) is the distance \(OP\). The particle is projected from the point \(C\) \((c, 0)\) with velocity \(nb\), in the direction which makes an angle \(\alpha\) with \(Ox\). Show that \(P\) moves on the ellipse \(b^2(x\sin\alpha - y\cos\alpha)^2 + c^2y^2 = b^2c^2\sin^2\alpha.\) Using \(t = \tan\alpha\) as parameter, or otherwise, show that all points of the plane which can be reached by projection from \(C\) with speed \(nb\) lie within or on the ellipse \(b^2x^2 + (b^2 + c^2) y^2 = b^2(b^2 + c^2).\)
A smooth hollow straight tube \(AB\) is inclined at a constant acute angle \(\alpha\) to the horizontal and is constrained to rotate with constant angular velocity \(\omega\) in a vertical plane through \(A\). A heavy particle is projected with speed \(u\) along the tube towards \(B\). Assuming that the tube is infinitely long, find the smallest value of \(u\) which will ensure that the particle never comes to rest relative to the tube. If \(u = \frac{g}{6}\omega^{-1}\sqrt{2}\), find how far the particle moves along the tube before coming to relative rest.
Two unequal masses, \(m_1\) and \(m_2\), are fixed to the ends of a light elastic spring of length \(k\). The spring is laid on a smooth horizontal table and compressed through a distance \(l\). Both ends are then released simultaneously. Investigate mathematically the subsequent motion of the system.
The motion of a yo-yo is represented in the following approximation. Two equal uniform heavy circular discs are fixed one on each end of a uniform short right cylinder. The centres of the discs being in line with the axis of the cylinder. The radius of gyration of the reel-shaped solid thus formed (the yo-yo) is \(k\). A light inelastic thread of negligible thickness is fixed to a point on the central section of the cylinder and the thread wound round the cylinder until length \(l\) has been wound on. The system is then held with the discs in vertical planes and the thread taut and vertical, being fixed at a point whose distance above the axis of the yo-yo is great compared with \(a\). The yo-yo is then released and falls. Assuming that the planes of the discs remain vertical and that the deviation of the unwound thread from the vertical can be neglected, find the time that elapses up to the instant when the thread is completely unwound. Describe qualitatively the motion subsequent to that instant, explaining in particular what happens to the downward momentum of the yo-yo.
Derive a formula for the area of a surface of revolution. An oblate spheroidal surface is formed by rotation of the ellipse \(x^2/a^2 + y^2/b^2 = 1\) about its minor axis. Prove that the area of the surface is \(2\pi a^2 \left( 1 + \frac{1 - e^2}{2e} \log_e \frac{1 + e}{1 - e} \right),\) where \(e\) is the eccentricity \([e = \sqrt{(1 - b^2/a^2)}]\). Given that \(e\) is small, find an approximate expression for this area as a sum of powers of \(e\), correct to \(O(e^4)\).
A moving point \(P\) describes a smooth plane curve, \(\Gamma\) with continuous gradient. The arc \(AP\) from a fixed point \(A\) on \(\Gamma\) has length \(s\). \(PP'\) is the tangent at \(P\) in the direction along which \(s'\) measures arc the curve from the fixed point \(A'\) on \(\Gamma\), and \(PP' = a\) (constant). The locus of \(P'\), as \(P\) moves, is \(\Gamma'\). \(C\) is the centre of curvature of \(\Gamma\) at \(P\) and \(C'\) is the centre of curvature of \(\Gamma'\) at \(P'\). Prove \(ds'/ds = R/\rho\), where \(R = CP\) and \(\rho = CP\). Prove also that \(C'\) lies on \(CP'\) and \(CC'/C'P' = -\frac{d\rho d\rho}{R^2 ds}\).