Interpret the expressions \(\displaystyle\int x \frac{dy}{ds} ds\) and \(\displaystyle\int y \frac{dx}{ds} ds\) when taken round the boundary of a closed plane curve. Trace the curve \[ y^4 - 2xy + x^3 = 0 \] and prove that the area of a loop is \(\frac{2}{5}\).
Forces \(P, Q, R\) acting at a point \(O\) are in equilibrium and a straight line meets their lines of action in \(A, B, C\) respectively; shew that, with certain conventions of sign, \[ \frac{P}{OA} + \frac{Q}{OB} + \frac{R}{OC} = 0. \]
Shew that any system of co-planar forces, not in equilibrium, may be reduced to a single force or a couple. \(D, E, F\) are the middle points of the sides \(BC, CA, AB\) respectively of a triangle \(ABC\). Forces of 1, 2, 3 lbs. act along \(BC, CA, AB\) and forces of 2, 3, 5 lbs. act along \(FE, ED, DF\). Find the magnitude of the resultant force, the inclination of its line of action to \(BC\) and its perpendicular distance from \(A\).
Two equal cylinders lie in contact on a horizontal plane and an isosceles triangular wedge is placed symmetrically upon them so as to touch each cylinder along a horizontal line. If the angle of the wedge is \(\alpha\) and the angle of friction is greater than \(\frac{1}{4}(\pi-\alpha)\), shew that the cylinders will not move, no matter however great the weight of the wedge may be.
[A diagram shows a simple truss A-C-B, with C above the line AB, and a vertical member from C to the midpoint of AB.] The figure represents a framework of rigid rods, supposed to be loosely jointed at their intersections, and to be of negligible weight. A given weight is suspended from \(C\) and the framework is kept in equilibrium by vertical forces at \(A\) and \(B\). Draw a stress diagram for the figure shewing the tensions and thrusts in the rods.
State the principle of virtual work, and explain how it may be applied to determine the unknown reactions of a system. A square \(ABCD\) formed of light rods, loosely jointed, has the side \(AB\) fixed. The middle points of \(AB, BC\) are joined by a string which is kept taut by a force \(P\) acting at the middle point of \(AD\) parallel to \(AB\). Shew that the tension of the string is equal to \(P\sqrt{2}\).
Two equal particles are connected by a string 5 feet long and lie close together at the edge of a window ledge 63 feet from the ground. One of them is pushed gently over the edge. Find the time it will take to reach the ground.
The energy of 1 lb. of powder is 75 foot-tons. Shew that the weight of charge necessary to produce an initial velocity of 1500 feet per second in a projectile weighing 600 lbs. is at least 125 lbs. (Neglect the recoil of the gun.)
A particle moves in a circle of radius \(a\) with constant angular velocity \(\omega\). Shew that the acceleration is directed towards the centre and is equal to \(a\omega^2\). An elastic string of unstretched length \(a\) is stretched by an amount \(b\) when it supports a certain mass at rest. Shew that when rotating steadily at the rate of \(n\) revolutions per second round the vertical through one end (the same mass being attached to the other end) the inclination to the vertical is \[ \cos^{-1}\left(\frac{1}{a}\left(\frac{g}{4\pi^2 n^2}-b\right)\right). \] Discuss this result when \(2\pi n > \sqrt{(g/b)}\).
A particle is projected from a point on the ground at the centre of a circular wall of radius \(a\) and height \(h\). Shew that the least velocity of projection which will enable the particle to clear the wall is \(\sqrt{g\{h+\sqrt{(a^2+h^2)}\}}\).