Three smooth heavy cylinders \(A, B, C\) lie on a table, with \(B\) between \(A\) and \(C\) and touching each of them. \(A\) and \(C\) have equal radii \(a\) and \(B\) has weight \(W\) and radius \(b(
Prove that couples in one plane and of equal and opposite moment are in equilibrium. The ends of a rod are constrained by smooth rings to slide on two horizontal fixed rods which meet at \(A\). To the centre of the rod is attached a string which passes over a smooth pulley at \(A\) and is fastened to a weight which hangs freely. A couple is applied to the rod in a horizontal plane. Show that in any position of equilibrium of the rod the couple applied is proportional to the cosine of the angle which the string makes with the rod.
Explain the term `cone of friction.' The figure shows a log of square section \(ABCD\) split along a plane \(EF\) parallel to \(BC\) and resting in equilibrium upon two smooth horizontal parallel rails on the same level, so that \(AC\) is vertical. Show that the coefficient of friction between the two faces \(EF\) must not be less than \(BE/EA\). [Diagram of a square ABCD, viewed in perspective, tilted so that A is the highest point and C is the lowest. E is a point on AB and F is a point on CD. A line connects E and F.]
Obtain the equations of equilibrium of a rigid lamina by applying the principle of virtual work. A light rhombus formed of rods smoothly jointed at \(A, B, C, D\) rests in a vertical plane with \(A\) vertically above \(C\) and the rods \(AB, AD\) over smooth pegs at the same level at a distance \(2c\) apart. \(B, D\) are connected by a light rod so that the angle \(A\) of the rhombus is \(2\alpha\). Show that if a weight \(W\) is hung from \(C\), the stress in the rod \(BD\) is \(W \left(\frac{c}{2a \sec\alpha \operatorname{cosec}^2\alpha} - \tan\alpha\right)\), \(a\) being a side of the rhombus; and find the condition that the stress may be a tension.
Four equal light rods \(AB, BC, CD, DE\) have smooth hinges at \(B, C, D\) and the centres of \(AB\) and \(DE\) are hinged to the ends of a light rod. Equal weights are hung from \(B, C, D\) and the system is supported by vertical strings at \(A\) and \(E\). Show by a force diagram that if \(\alpha, \beta\) are the inclinations of \(AB\) and \(BC\) to the horizontal, \(\tan\alpha = 6\tan\beta\).
Given the resolved parts of a velocity in two directions, find the velocity by geometrical construction. A vessel steams at uniform speed in a steady wind on two given courses, and the angle made by the trail of smoke with the course is observed in each case. Find, by geometrical construction, the direction of the wind.
A train of forty waggons, each of 10 tons, is drawn up an incline of 1 in 100 by an engine of 100 tons in front, aided by an engine of 60 tons pushing behind. The load on the driving wheels of each engine is half its weight and the coefficient of friction between the wheels and rails is 0.14. Neglecting axle friction, the inertia of the wheels and the resistance of the air, find the greatest acceleration of the train possible and with this acceleration find the tension of the coupling attached to the front engine. How many couplings are slack?
Find the range of a gun on an inclined plane on which the gun is fixed, when the gun is pointed in a given direction. The enemy is known to be up the line of greatest slope through the gun and is within range but invisible from the gun. An aeroplane is sent out to scout and gives a signal when it is vertically above the enemy. The gun is kept pointed at the aeroplane and is fired when the signal is given. Show that the enemy will be hit, provided the aeroplane flies in a certain vertical circle. From which part of the circle should the signal be given according as the enemy is entrenched or not?
Find the kinetic energy lost in the impact of two smooth balls. Find the angle through which the direction of motion of a ball \(A\) is turned by striking an equal ball \(B\) at rest, and prove that, if \(e=\frac{2}{3}\) and the direction of motion of the centre of \(A\) before impact is tangential to \(B\), the angle is about 50\(^{\circ}\).
A particle describes a circle with variable speed. Find the tangential and normal components of the force on the particle. \(AB\) is the upper side (\(a\)) of the square cross-section of a log which has two sides of the section vertical. A particle is attached to \(A\) by a string, of length \(l(>4a)\), which is initially stretched out along \(BA\) produced. Prove that, if the particle is projected downwards with velocity greater than \(\{g(3l-8a)\}^{\frac{1}{2}}\), the string will wrap tightly round the log till the particle strikes the log.