Show that a system of forces acting in a plane can be reduced to two forces of which one acts at a given point and the other acts in a given line. If referred to rectangular axes the original forces are \((X_1, Y_1), (X_2, Y_2), \dots, (X_n, Y_n)\) acting at \((x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)\) respectively, effect the reduction
A uniform heavy rod \(AB\) of length \(2a\) is in equilibrium in a horizontal position in contact with a rough plane of coefficient of friction \(\mu\) inclined to the horizontal at an angle \(\alpha\). At the end \(A\) a gradually increasing force is applied up the plane in the direction of the line of greatest slope through \(A\). Show that when the rod begins to move it turns about a point at distance \[ a\left(2-\frac{2\tan\alpha}{\mu}\right)^{\frac{1}{2}} \] from \(A\).
A uniform beam of length \(2a\) and weight \(2wa\) rests on two supports at the same horizontal level at equal distances \(a-b\) from its ends. Find the ratio \(b\) must have to \(a\) in order that the greatest absolute value of the bending moment shall be as small as possible. Show that for this ratio, the bending moments at the centre and the supports are numerically equal and of amount \(wa^2(\frac{3}{4}-\sqrt{2})\).
A heavy flexible chain hanging in equilibrium between two fixed points is so constructed that the weight per unit length at any point is proportional to the tension at that point. Prove that, referred to certain rectangular axes, the equation of the curve in which the chain hangs may be put in the form \[ y=\log\sec x \] when the unit of length is suitably chosen.
A particle projected vertically upwards under gravity in a resisting medium that produces a retardation proportional to the velocity reaches a greatest height \(h\) in time \(T\). If \(h_0\) and \(T_0\) are the corresponding greatest height and time for projection with the same speed but in the absence of resistance, prove that \(h_0> h\) and \(T_0< T\).
A uniform rod is moving in a plane in a direction at right angles to its length when it collides with an imperfectly elastic particle of equal mass at rest in the plane but free to move. Show that, whatever the value of the coefficient of restitution, the angular velocity imparted to the rod is greatest if the point of impact is at distance \(k\sqrt{2}\) from the centre, where \(k\) is the radius of gyration of the rod about its centre. Find the maximum angular velocity in terms of \(l\) the length of the rod, \(V\) its velocity just before impact, and \(e\) the coefficient of restitution.
A particle can be projected with fixed speed \(V\) from a given point \(O\) of a plane inclined to the horizontal at an angle \(\alpha\). Prove that the area within range is an ellipse, with \(O\) as focus and of area \(\pi V^4/g^2 \cos^3\alpha\), where \(g\) is the acceleration of gravity.
An inextensible cord is being unwound from a flat circular reel of centre \(O\). The radius \(OC\) to the point of contact rotates at constant angular velocity \(\omega\), and the unwound portion \(CD\) is kept straight and in the plane of the reel. Prove that the acceleration of the end \(D\) is always towards a certain point \(B\) at a fixed distance from \(C\) in \(OC\) produced and of magnitude \(\omega^2.DB\).
A heavy ring of mass \(2m\) can slide on a fixed smooth vertical rod and is attached to one end of a light inextensible string that passes over a small smooth peg at perpendicular distance \(a\) from the rod. The string carries at its other end a heavy particle of mass \(4m\) that hangs freely. If the system is released from rest with the ring at the level of the peg, prove that the ring descends a distance \(\frac{8}{3}a\) before coming to rest again. If at this instant the mass of the ring is suddenly reduced to \(m\), show that it will slide upwards through a distance \(\frac{12}{5}a\) before coming to rest again.
A uniform heavy rod \(AB\) hangs in equilibrium by two equal inextensible strings \(OA, OB\) attached to a fixed point \(O\). If one of the strings is suddenly cut, show that the tension in the other is instantaneously reduced in the ratio \[ 2/(\operatorname{cosec}^2\theta+3), \] where \(\theta\) is the angle \(OAB\).