Prove that the line drawn through any point of the parabola \(y^2=4ax\) at right angles to the line joining the point to the vertex is normal to a fixed parabola whose equation is of the form \(y^2=16a(x+4a)\).
The line \(y=k\) cuts the ellipse \(b^2x^2+a^2y^2=a^2b^2\) in \(K\) and \(K'\); through these points any parallel lines \(KP, K'P'\) are drawn cutting the ellipse in \(P\) and \(P'\). Prove that the locus of the pole of \(PP'\) is a similar co-axal ellipse.
Prove that, if \(P\) and \(Q\) are points on the cardioid \(r=a(1+\cos\theta)\) such that the angle between the tangents at \(P\) and \(Q = \alpha\), the chord \(PQ\) subtends an angle \(\frac{1}{2}(\pi-\alpha)\) at the cusp.
A frame consists of seven light rods jointed to form three equilateral triangles \(ABC, BCD, CDE\). The frame rests on smooth vertical supports at \(A\) and \(E\), with \(ACE\) and \(BD\) horizontal, \(BD\) being above \(AE\), and carries loads of 12 cwt. at \(B\) and 10 cwt. at \(C\). Determine the stresses in the rods, stating which are in tension and which in compression.
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.