A heavy particle is attached to a fixed point by a fine inextensible string of length \(a\), and is projected horizontally with velocity such that the string becomes slack when inclined at angle \(\theta\) to the upward drawn vertical. Shew that, when the string again becomes taut, it makes angle \(3\theta\) with the vertical, that the time of the free parabolic path is \(4 \sin\theta \sqrt{\frac{a}{g}\cos\theta}\) and that the velocity of the particle immediately after the impulse is \(\frac{1}{2}(2\cos 2\theta - \cos 4\theta)\sqrt{(ga \cos\theta)}\).
Two opposite sides of a quadrilateral inscribable in a circle lie respectively along the coordinate axes \(Ox, Oy\). If the diagonals of the quadrilateral intersect in a given point, shew that the locus of the centres of the circles is a straight line.
Trace the curve \(y^2(a+x) = x^2(3a-x)\), and shew that the area of the loop and the area included between the curve and the asymptote are both equal to \(3\sqrt{3}a^2\).
A mass \(M\) suspended at the end of a vertical spring oscillates harmonically with amplitude \(a\). At the moment of the greatest velocity of \(M\) a mass \(m\) at rest is placed upon it and remains upon it; shew that the resulting amplitude of oscillation is \(\sqrt{\left(\mu^2\frac{M}{M+m} + \frac{m^2}{(M+m)^2}\right)a^2}\), where \(\mu\) is the mass which suspended alone at the end of the spring stretches it unit distance.
A rectangular cistern to contain 1 cubic yard is constructed so that the whole surface of sides and bottom may be as small as possible; prove that the bottom is a square and that the depth is approximately \(22\frac{1}{2}\) inches.
Find \(\int \frac{dx}{x(1+x+x^2)}\), \(\int \frac{\sqrt{a^2-x^2}}{x^2}dx\), \(\int \frac{dx}{\sin x}\). The thickness of a circular disc of radius \(a\) at a distance \(r\) from the centre is \(2ap/(4a^2-r^2)^{\frac{1}{2}}\); find the average thickness of the disc.
Two forces \(P, Q\) of given magnitude act at fixed points \(A, B\). Their lines of action are in a fixed plane through \(AB\), and are always at right angles to one another, but can turn about \(A, B\) respectively through any equal angles in the same sense. Prove that their resultant passes through a fixed point \(O\), whose distance from \(AB\) is equal to \[ \frac{P \cdot Q}{P^2+Q^2} AB. \] If \(P, Q\) are interchanged, so that the resultant now goes through a different fixed point \(O'\), prove that \[ OO' = \frac{P^2-Q^2}{P^2+Q^2} AB. \]
The diagram represents a system of seven light rods smoothly jointed at \(A, B, C, D, E,\) and supporting a load \(W\) at \(C\). \(A\) is fixed and \(D\) rests against a smooth support, the horizontal and vertical rods being all of equal length. [The diagram shows a structure. A is a point. AB is horizontal to the right. BC is horizontal to the right, continuing AB. AE is vertical downwards. ED is horizontal to the right. B is connected to E. C is connected to E. A load W hangs from C. The lengths AB, BC, ED, AE are equal. D is against a vertical wall.] Construct a force-diagram shewing the reaction in each rod, and state in each case whether it is a tension or a thrust. Find also the pressure at \(D\), and the resultant action at \(A\).
A hollow triangular prism with open ends is formed from three rectangular sheets of metal of uniform thickness. Its cross section is a triangle \(ABC\), obtuse-angled at \(C\). Prove that, if placed with the face \(BC\) in contact with a horizontal plane, it will topple over if \[ 2a^2 < (c-b)(b+c-a). \]
Two beads \(A, B\), whose weights are \(w_1, w_2\) are tied to the ends of a string, on which is threaded a third bead \(C\) of weight \(W\). The beads \(A, B\) can slide on a rough horizontal rod, whose coefficient of friction with each bead is \(\mu\). If, when \(A, B\) are as far apart as possible, the strings \(AC, BC\) each make an angle \(\theta\) with the vertical, prove that, if \(w_1>w_2\), \[ \tan\theta = \mu \{1+2(w_2/W)\}. \]