Problems

Consider three skew lines, \(a, b\) and \(c\), in space. \(A_1, A_2, A_3\) and \(A_4\) are four points on the line \(a\), and \(l_1, l_2, l_3\) and \(l_4\) are the lines through these points which meet \(b\) and \(c\). Shew that the cross ratio \((A_1 A_2 A_3 A_4)\) is equal to that of the four planes through \(a\) which contain the lines \(l_1, l_2, l_3\) and \(l_4\). (The cross ratio of four planes through a line \(\lambda\) is the cross ratio of the four points in which the planes meet any line which does not meet \(\lambda\).) By considering homographic ranges on \(a\), or otherwise, shew that, if \(d\) is another line, there are two lines which meet \(a, b, c\) and \(d\).

Obtain the equations of the two parabolas which pass through the points \((0,0), (7,0), (0,5),\) and \((3,-1)\), and the equations of their axes.

Shew that the orthocentre of the triangle formed by tangents to the parabola \(y^2 = 4ax\) at the points \((at_1^2, 2at_1), (at_2^2, 2at_2)\) and \((at_3^2, 2at_3)\) is the point \(\{-a, a(t_1+t_2+t_3) + at_1t_2t_3\}\). Normals are drawn to the parabola \(y^2=4ax\cos\alpha\) from any point on the line \(y=b\sin\alpha\). Shew that the orthocentre of the triangle formed by the tangents at the feet of the normals lies on the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

Four equal uniform freely jointed rods, forming a rhombus, rest in equilibrium with one diagonal vertical and the two lower rods supported on two smooth pegs at the same horizontal level. If \(a\) is the length of each rod, \(2c\) the distance between the pegs, and \(2b\) the horizontal diagonal of the rhombus, prove that \(b^3=a^2c\). If there is no reaction at the lowest joint, prove that \(8a=5\sqrt{5}c\).

Two rough uniform cylinders of equal radius rest in contact, with their axes horizontal, on a plane inclined at an angle \(\alpha\) to the horizon. If \(W_1\) is the weight of the upper cylinder and \(W_2\) of the lower, prove that \(W_1>W_2\), and that the coefficient of friction between the cylinders exceeds \((W_1+W_2)/(W_1-W_2)\). If \(\mu\) is the coefficient of friction between the plane and the upper cylinder, prove that \[ \tan\alpha < \frac{2\mu}{\mu+1} \cdot \frac{W_1}{W_1+W_2}. \]

A uniform thin hollow hemispherical bowl is in equilibrium on a horizontal plane with a smooth uniform straight rod resting partly within and partly without it. If the weight of the rod is half that of the bowl and its length is equal to the diameter of the spherical surface, prove that the inclination of the rod to the horizon is \[ \tan^{-1}\frac{1}{2}(2-\sqrt{2}). \]

Two straight rods passing through the fixed points \(A\) and \(B\) revolve uniformly in one plane about these points in the same direction, the one through \(B\) three times as fast as the one through \(A\). The rods, starting in the same direction \(AB\), intersect in \(P\) at any time, and \(N\) is the foot of the perpendicular from \(P\) on \(AB\). Prove that the motion of \(N\) is simple harmonic, with amplitude equal to \(AB\) and period equal to half the time of one revolution of the rod through \(A\).

A mass \(M\) lb. is to be raised through a vertical height \(h\) feet, starting from rest and coming to rest under gravity, by a chain whose tension is not allowed to exceed \(P\) lb. wt. Neglecting the weight of the chain, prove that the shortest time in which this can be done is \[ \sqrt{\frac{2hP}{g(P-M)}} \text{ seconds.} \] If a large amount of material is to be raised, prove that the total time occupied on the upward journeys will be as short as possible if the material is sent up in loads of \(\frac{2}{3}P\) lb.

In a smooth fixed circular tube, of radius \(a\) and small bore, in a vertical plane, are two particles of masses \(m\) and \(2m\), connected by a light inextensible string of length \(a\pi\). With the particles at the ends of the horizontal diameter, and the string in the upper half of the tube, the system is released from rest. Prove that, when each particle has described an arc \(a\theta \left[\theta < \frac{\pi}{2}\right]\), the pressure between the lighter particle \(m\) and the tube is \(\frac{1}{3}mg\sin\theta\), and the tension of the string is \(\frac{2}{3}mg\cos\theta\).

A gun barrel of mass 4 tons is attached to a rigid mounting through a hydraulic buffer, which exerts an opposing force of \(4+\frac{v}{2}\) tons weight when the barrel is recoiling at \(v\) feet per second, the force and the motion both being in the direction of the barrel, which has an elevation of \(30^\circ\). Prove that, when a shot of mass 1 cwt. is discharged at 1280 feet per second, the distance of recoil is 2.4 feet. [\(\log_e 5 = 1 \cdot 6\).]