A frame consists of nine light rods jointed together. AB is vertical, B being above A; the rods AC, BC and the rods AD, DB form equilateral triangles with AB; the rods CE, DF are perpendicular to AC, AD respectively; and the rods AE, AF are in the same horizontal line. The whole frame is supported in a vertical plane by supports at E and F, and carries loads of 20 cwt. at C and 30 cwt. at D. Determine the stresses in the rods, stating which are in compression and which in tension.
The engine of a train of 300 tons can just attain a speed of 60 miles per hour on the level. Assuming that the resistance varies as the square of the velocity and that the horse-power is constant and equal to 1000 units, shew that the train starting from rest will attain a speed of 30 miles per hour after moving through a space of 386 feet approximately. [\(\log_e \frac{4}{3}=0.1335\).]
Two particles m and m' are connected by a string of length \(l\) and rest on a smooth horizontal table with the string taut. If m is just at the edge of the table and is allowed to fall, determine the motion of m and m' after m' is drawn over the edge of the table, and shew that the system will rotate with uniform angular velocity \(\sqrt{\{2gm/l(m+m')\}}\).
Given two points \(A, B\), prove the existence of a system of circles with the property that the tangents from any point \(P\) on the perpendicular bisector of \(AB\) to the circles of this system are equal to \(PA\). Give a geometrical construction for the circles of the system, which (i) touch a given line, (ii) pass through a given point \(P\). Prove that, if the circle of the system through \(P\) cuts \(AP\) again in \(P'\), \(AB\) is one of the angle bisectors of \(PBP'\), and that, if the locus of \(P\) is a straight line, the locus of \(P'\) is another straight line.
If \(s=0\) is the equation of a conic, \(t=0\) the equation of one of its tangents and \(p=0\) the equation of one of its chords, interpret the equations \[ s+kpt=0, \quad s+kt^2=0, \quad s+kt=0, \] where \(k\) is a parameter. \(PP'\) is a chord of a conic \(S_o\); a conic \(S\) passes through \(P\) and has three point contact with \(S_o\) at \(P'\), and another conic \(S'\) passes through \(P'\) and has three point contact with \(S_o\) at \(P\). Prove that the other chord of intersection of \(S\) and \(S'\) is concurrent with the tangents at \(P\) and \(P'\). The tangents at \(B, C\) to a conic meet in \(A\) and the tangents at \(B', C'\) meet in \(A'\); prove that there is another conic which touches \(A'B', A'C'\) where they are cut by \(BC\), and also touches \(AB, AC\) where they are cut by \(B'C'\).
Find the conditions that the roots of \[ x^3-ax^2+bx-c=0 \] shall be (i) in G.P., (ii) in A.P., (iii) in H.P. Show that if the roots are not in A.P. then there are in general three transformations of the form \(x=y+\lambda\) such that the transformed cubic in \(y\) has its roots in G.P.
From the ordinary geometrical definitions of \(\sin x, \cos x\) and the assumption that \(\frac{d}{dx}(\sin x) = \cos x\), deduce that, if \(x\) is positive, \[ \cos x - 1 + \frac{x^2}{2!} - \dots - (-1)^m \frac{x^{2m}}{(2m)!} \] and \[ \sin x - x + \frac{x^3}{3!} - \dots - (-1)^m \frac{x^{2m+1}}{(2m+1)!} \] are positive or negative according as \(m\) is odd or even. Prove also that, if \(x^2<1\), one value of \(\tan^{-1}x\) lies between \[ x-\frac{1}{3}x^3 \quad \text{and} \quad x-\frac{1}{3}x^3+\frac{1}{5}x^5.\]
The area \(S\) and the semi-perimeter \(s\) of a triangle are fixed. Prove that for one of the sides \(a\) to be a maximum or a minimum it must be a root of the equation \[ s(x-s)x^2+4S^2=0.\] Hence show that there is one maximum and one minimum.
Two rough planes intersect at right angles in a horizontal line and make angles \(\alpha, \frac{\pi}{2}-\alpha\) (\(0 < \alpha < \frac{\pi}{4}\)) with the horizontal. Two equal rough cylinders with their axes in the same horizontal plane rest in contact with each other and each in contact with one plane. Prove that, if all the surfaces are equally rough, the coefficient of friction is not less than \[ \frac{\cos 2\alpha}{\sin\alpha+\cos\alpha+\sin 2\alpha}.\]
A circular disc of weight \(w\) and radius \(a\) can slide on a smooth vertical rod passing through a small hole at its centre. It is supported horizontally by \(n\) light vertical rods each of length \(2a\) freely hinged to it at equal intervals round its circumference, the upper ends of the rods being freely hinged to fixed supports. An elastic band of modulus \(\lambda\) and natural length \(c\) is placed round the rods. Prove that the equilibrium is stable if \[ w > \lambda n \sin\frac{\pi}{n} \left( \frac{2a}{c} n \sin\frac{\pi}{n} - 1 \right), \] and that otherwise there will be stable equilibrium when the rods are inclined to the vertical at an angle \[ \cos^{-1} \frac{c(w+\lambda n \sin\frac{\pi}{n})}{2a\lambda n^2 \sin^2\frac{\pi}{n}}.\] Assume that the band remains taut throughout and neglect friction.