A system of coplanar forces acts on a rigid body, and \(A, B, C, D\), are four points in the plane of the forces. If the moments of the system about the four points are denoted by \(\alpha, \beta, \gamma, \delta\) respectively, prove that \[ a\alpha - b\beta + c\gamma - d\delta = 0, \] where \(a\) denotes the area of the triangle \(BCD\), \(b\) of \(CDA\), \(c\) of \(DAB\), and \(d\) of \(ABC\). (The areas of triangles in different senses are given opposite signs.) In connexion with this theorem
Three similar uniform rods \(AB, BC, CD\) are freely hinged together at \(B\) and \(C\), and \(A, D\) are attached to light rings which slide on a rough horizontal rail. If the coefficient of friction between a ring and the rail is \(\mu\), shew that the greatest inclination that \(AB\) and \(CD\) can make to the vertical in a symmetrical position of equilibrium is \(\phi\), where \(\tan\phi = \frac{3}{2}\mu\). If the joints at \(B\) and \(C\) are not smooth but can sustain a frictional couple \(\gamma\), shew that, if \(\gamma\) is not too large, the greatest inclination is increased by \(\sin^{-1}\frac{\gamma}{aW\sqrt{9\mu^2+4}}\), where \(W\) is the weight and \(2a\) the length of a rod.
Explain what is meant by the potential energy of a dynamical system on which only conservative forces act. Shew that the potential energy is stationary in a position of equilibrium, and that if the potential energy is a minimum the equilibrium is stable. A bead slides on a smooth circular hoop fixed in a vertical plane, and is attached to a particle, whose mass is equal to that of the bead, by a light inelastic string passing through a smooth ring placed very near to the highest point of the hoop. Determine the positions of equilibrium, and discuss their stability.
A particle is projected in a given vertical plane from a point \(O\), the horizontal and vertical components of velocity being \(u\) and \(v\) respectively. If \(u, v\) are connected by a relation \[ au^2 + v^2 = 2gh, \] where \(a, h\) are positive constants, shew that the envelope of the trajectories is a parabola whose vertex is at a height \(h\) above \(O\) and whose latus rectum is \(4h/a\). Shew also that to reach the point \(Q\) on the envelope the elevation of the direction of projection must be \(\tan^{-1}(2h/x)\), where \(x\) denotes the horizontal projection of \(OQ\). A gun is mounted on a truck and can fire a shell of mass \(m\) in a vertical plane parallel to the rails, the mass of the gun and truck together being \(M\). Find the envelope of the trajectories (i) on the assumption that the velocity of the shell relative to the gun is constant and equal to \(\sqrt{2gh}\), (ii) on the assumption that the total kinetic energy immediately after the shell leaves the gun is constant and equal to \(mgh\).
Explain briefly the principles of conservation of momentum and energy, and apply them to the solution of the following problem. A bead of mass \(M\) slides on a smooth straight horizontal wire, and a light rod of length \(l\) is freely attached to the bead and carries a particle of mass \(m\) at the other end. The rod is held in the vertical plane through the wire at an inclination \(\alpha\) to the vertical, and released. Shew that, if the inclination of the rod to the vertical at a subsequent time is \(\theta\), then \[ (M+m\sin^2\theta)l\dot{\theta}^2 = 2(M+m)g(\cos\theta - \cos\alpha). \]
If \(\alpha, \beta, \gamma\) be the distances of the centre of the nine-point circle from the vertices of a triangle, \(p\) its distance from the orthocentre, and \(R\) the radius of the circumscribing circle, then \[ \alpha^2+\beta^2+\gamma^2+p^2 = 3R^2. \]
Show that the equation \[ \{a(x^2-c)-b(y^2-c)\}^2 + 4\{axy+h(y^2-c)\}\{bxy+h(x^2-c)\}=0 \] represents four lines forming the sides of a rhombus.
A rectangular plate has sides ten inches and five inches. If equal squares are cut out at the four corners and if the sides are then turned up so as to form an open box, find the volume of the greatest box so formed.
Trace the curve \[ x = 2a \sin^2 t \cos 2t, \quad y = 2a \sin^2 t \sin 2t. \] Show that the length of its arc is \(8a\), and find the radius of curvature at the origin.
Show that if \[ I_n = \int_0^\infty \frac{dx}{(1+x^2)^n}, \] where \(n\) is a positive integer, then \[ I_{n+1} = \left(1-\frac{1}{2n}\right)I_n. \] and hence evaluate \(I_n\).