Two equal rectangular blocks of length \(a\) having square ends of side \(b\) are placed on a horizontal table with two square faces in contact, and a third block of the same size is placed symmetrically on top of them. Equal forces are then applied to the centres of the end faces of the lower blocks. Prove that, provided the horizontal components of these forces are greater than \(3Wa/2b\), the table may be removed without disturbing equilibrium, \(W\) being the weight of each block.
A sphere of mass \(M\) supported by a vertical inextensible string is struck by a sphere of mass \(m\) which is falling vertically with velocity \(v\), the line joining the centres of the spheres being inclined at an angle \(\alpha\) to the vertical at the instant of impact. Prove that the loss of energy \[ =\frac{1}{2}Mmv^2(1-e^2)\cos^2\alpha/(M+m\sin^2\alpha), \] where \(e\) is the coefficient of restitution for the spheres.
Two masses \(m,m'\), connected by a weightless rod, lie on a smooth horizontal table. The rod is struck at right angles to its length by an impulsive force \(F\); find the velocities of the masses, and show that the kinetic energy is least if \(F\) is applied at the centre of gravity of the masses.
Give a summary account of the relations between the fundamental principles of Rigid Statics and Rigid Dynamics, referring especially to d'Alembert's Principle and to the principle of Transmissibility of Force.
Show that a particle moving under the action of a fixed centre of gravitation describes a conic. Show that in an orbit of period \(T\) and small eccentricity \(e\) the polar angle \(\theta\) is given as a function of the time by the equation \[ \theta = 2\pi t/T + 2e\sin 2\pi t/T. \]
A rigid body moves about a fixed point under the action of no forces except the reaction at the fixed point. Show that its motion may be described by saying that its momental ellipsoid rolls without slipping on a fixed plane, and show that its component angular velocity in a direction perpendicular to this plane is constant. Show also that if, with the usual notation, the surface \[ Ax^2+By^2+Cz^2=M(x^2+y^2+z^2)^2 \] be traced in the body, it will roll throughout the motion on a fixed sphere.
Show that the equation of the curve taken by a uniform chain hanging freely under gravity is of the form \(y=c\cosh x/c\), that the arc-length measured from the vertex is \(c\sinh x/c\), and that the tension at any point is proportional to its height above the directrix. A uniform chain hangs symmetrically with its ends fixed a distance \(4a\) apart and its middle supported at the same level as its ends by a smooth peg. Show that the arrangement is stable provided the length of the chain is less than \(4c\sinh a/c\), where \(c\) is the positive root of the equation \(a/c = \coth a/c\).
Obtain the equations for the free motion of a particle relative to the surface of the rotating earth, near a place whose North latitude is \(\lambda\), in the approximate form \begin{align*} \ddot{x}-2\omega\dot{y}\sin\lambda &= 0, \\ \ddot{y}+2\omega\dot{x}\sin\lambda+2\omega\dot{z}\cos\lambda &= 0, \\ \ddot{z}-2\omega\dot{y}\cos\lambda+g &= 0, \end{align*} where terms of the order \(\omega^2x, \omega^2y, \omega^2z\) are neglected; the axes are rectangular and \(Oz\) is along the upward vertical (the vertical being the direction of apparent gravity), \(Oy\) Easterly and \(Ox\) Southerly; \(\omega\) is the earth's angular velocity and \(g\) the acceleration of apparent gravity. A particle is projected in a Northerly direction at an angle \(\alpha\) with the horizontal. Show that when it again reaches the level of projection it will have been deflected to the East or West according as \(\tan\alpha\) is less than or greater than \(3\tan\lambda\).
A homogeneous circular cone of mass \(M\) rolls with the rim of its base (radius \(R\)) on a rough horizontal plane and its vertex in contact with a fixed vertical cylinder of height \(h\) above the plane. Its axis always passes through a fixed point \(O\) on the axis of the cylinder and the motion is steady, the point of contact \(P\) with the plane describing a circle of radius \(r\) with angular velocity \(\omega\). Show that the reaction \(R\) at the vertex is given by \[ Rb = Mgc - M\omega^2(r-c)(r\cot\theta+a\csc\theta)+(A-C)\omega^2\sin\theta\cos\theta - C\omega^2 r\sin\theta/b, \] where \(c\) is the horizontal projection of \(GP\) (\(G\) being the centre of mass of the cone), \(\theta\) is the inclination of its axis to the vertical, and \(C,A\) are its principal moments at \(O\).
The point of suspension \(O\) of a rigid pendulum is given a very rapid simple harmonic vertical oscillation of period \(2\pi\tau\) and very small amplitude \(a\), regarded as of the same order of smallness as \(\tau\). Find the equation of motion for small deviations in a vertical plane from the position in which the centre of mass \(G\) is vertically above \(O\), and show that it has, to the first order in \(\tau\), a solution of the form \[ \theta=(Ae^{pt}+Be^{-pt})(1+q\tau\cos t/\tau+r\tau\sin t/\tau), \] where \(\theta\) is the inclination of \(OG\) to the vertical, \(p,q,r\) are definite finite constants, and \(A,B\) are the arbitrary constants of integration. Hence show that the vertical position is stable if \(a^2/\tau^2 > 2gl\), where \(g\) is the acceleration due to gravity and \(l\) is the length of the equivalent simple pendulum.