Six uniform heavy rods \(AB, BC, CD, DE, EF, FG\), each of length \(2a\) and weight \(W\), are freely jointed together to form a chain. \(C\) is joined to \(E\) by a light string of length \(\sqrt{2}a\), and \(B\) is joined to \(F\) by a light string of length \(4a/\sqrt{3}\). The ends \(A, G\) are freely hinged to small supports which are held together and slowly moved apart at the same level. Shew that the string \(CE\) is the first to tighten, and that this happens when the length \(AG\) is approximately equal to \(3.96a\).
Each of three particles \(A, B, C\) has a mass \(m\), and \(A\) is joined to \(B\), and \(B\) to \(C\) by similar light springs of natural length \(a\). The particles move in a straight line under no forces save the tensions of the springs. Shew that if the lengths of \(AB, BC\) respectively at time \(t\) are denoted by \(a+x, a+y\) respectively, then \[ \frac{d^2u}{dt^2} + n^2u = 0, \quad \frac{d^2v}{dt^2} + 3n^2v = 0, \] where \(u=x+y, v=x-y\), and \(amn^2\) is the tension required to double the length of either spring. Hence determine the length of \(AB\) at any time if the system, originally at rest with the springs unstretched, is set in motion by an impulse \(I\) on the particle \(C\) in the direction \(AC\).
A point moves in a circle of radius \(a\). If the radius through the point at time \(t\) makes an angle \(\theta\) with a fixed radius, shew that the acceleration of the point has components \(a\ddot{\theta}\) tangentially and \(a\dot{\theta}^2\) towards the centre of the circle. A light rod \(PQ\), of length \(b\), has a massive particle attached at \(Q\). The rod rests on a smooth table when the end \(P\) is seized and moved off in a horizontal circle of radius \(a\) with constant velocity \(a\omega\), the initial position of the rod being outside the circle and in line with the centre \(O\). In the subsequent motion the angle which \(PQ\) makes with \(OP\) produced is denoted by \(\phi\); shew that \[ b\dot{\phi}^2 = (a^2+b^2+2ab\cos\phi)\omega^2. \] Shew further that if \(a=b\), then \(\phi \to \pi\) as \(t\to\infty\).
Show that angles in the same segment of a circle are equal. A rod \(PQ\) slides with its ends \(P, Q\) on the two straight arms of a bent rod. At each position of \(P\) and \(Q\) lines \(PR, QR\) are drawn perpendicular respectively to the arms on which \(P\) and \(Q\) move. Show that, when the bent rod is fixed and \(PQ\) moves, the locus of \(R\) is a circle, and that, when \(PQ\) is fixed and the bent rod is moved, the locus of \(R\) is again a circle, of radius half the former circle and touching it at \(R\).
Prove that the number of combinations of \(n\) things \(r\) at a time is \(n!/\{r!(n-r)!\}\). A pack of cards is dealt (in the usual way) to four players. One player has just 5 cards of a particular suit; prove that the chance that his partner has the remaining 8 cards of that suit is \(1/(4.17.19.37)\).
Small errors \(\delta a, \delta b, \delta c\) are made in measuring the sides of a triangle; prove that the consequent error in reckoning the radius of the circumcircle is \[ \frac{1}{2}\cot A \cot B \cot C\left(\frac{\delta a}{\cos A} + \frac{\delta b}{\cos B} + \frac{\delta c}{\cos C}\right). \]
Prove that the radius of curvature of a curve, which is given by the equations \[ x=\phi(t), y=\psi(t), \] is \[ \frac{\left\{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2\right\}^{\frac{3}{2}}}{\left\{\frac{d^2y}{dt^2}\frac{dx}{dt} - \frac{d^2x}{dt^2}\frac{dy}{dt}\right\}}, \] and find the radii of curvature at the origin of the two branches of the curve given by the equations \[ y=t-t^3, \quad x=1-t^2. \]
Trace the curve \[ y^2(a+x) = x^2(3a-x), \] and show that the area of the loop and the area included between the curve and the asymptote are both equal to \(3\sqrt{3}a^2\).
Show that the series \[ 1 - \frac{1}{1+x} + \frac{1}{2} - \frac{1}{2+x} + \frac{1}{3} - \frac{1}{3+x} + \dots \] is convergent, provided that \(x\) is not a negative integer.
Two equal heavy cylinders of radius \(a\) are placed in contact in a smooth fixed cylinder of radius \(b\) (\(>2a\)); a third equal cylinder is placed gently on top of them, the axes of all the cylinders being horizontal. Show that the two lower cylinders will not separate if \[ b < a(1+2/\sqrt{7}). \]