Four uniform rods freely jointed form a parallelogram \(ABCD\), the weights of the opposite sides being equal. A weightless rod connects \(B\) and \(D\), and the system is suspended freely from \(A\). Prove that the stress in the rod \(BD\) is \(\frac{1}{2}W \cdot BD/AC\), where \(W\) is half the weight of the system.
State the laws of limiting friction. A uniform rod \(AB\) of weight \(W\) rests with one end \(A\) on a rough horizontal plane and the other end against an equally rough vertical wall, the vertical plane through the rod being perpendicular to the wall. Show that for equilibrium the inclination \(\alpha\) of the rod to the horizontal must be greater than \(\pi/2-2\epsilon\), where \(\epsilon\) is the angle of friction. If \(\alpha\) is less than \(\pi/2-2\epsilon\), show that the least force acting on \(A\) which will just maintain equilibrium is \[ W \cos(\alpha+2\epsilon)/2\sin(\alpha+\epsilon), \] and find the direction of this force.
A string passing over a smooth pulley carries a mass \(4m\) at one end and a pulley of mass \(m\) at the other. A string carrying masses \(m\) and \(2m\) at its ends passes over the latter pulley. Find the acceleration of the mass \(4m\) when the system is moving freely under gravity.
A particle is projected with a given velocity \(v\) from the foot of an inclined plane of slope \(\alpha\). The direction of projection lies in a plane containing the line of greatest slope and makes an angle \(\theta\) with the face of the plane. Prove that if the particle strikes the plane perpendicularly \(\cot\theta = 2\tan\alpha\). Show that, for different values of \(\alpha\), the range on the plane when the particle strikes it perpendicularly cannot be greater than \(v^2/g\sqrt{3}\).
Two smooth rings \(A, B\), each of mass \(m\), can slide on a smooth horizontal wire; a light string \(ACB\) has its ends attached to the rings and has an equal mass \(m\) attached to it at its middle point \(C\). The rings \(A, B\) are released from rest when the angle \(ACB\) is \(60^\circ\). Prove that the mass \(C\) begins to descend with an acceleration \(g/7\).
Given the motion of two smooth spheres before impact, write down equations to determine their motion after direct impact. A smooth inclined plane of slope \(\alpha\) and mass \(M\) is free to move on a smooth horizontal plane in a direction perpendicular to its edge. A spherical ball of mass \(m\) is dropped on it. Prove that the ball will rebound in a direction inclined to the horizontal at an angle \[ \tan^{-1}\left\{\frac{(M+m)\sin^2\alpha - Me\cos^2\alpha}{M(1+e)\sin\alpha\cos\alpha}\right\}, \] where \(e\) is the coefficient of restitution.
Reduce a system of given coplanar forces to a force or a couple. A, B, C, D are successive corners of a square whose side is 100 inches in length. Forces 2, 3, 1, 5, 4 lbs. act respectively along AB, BC, DC, DA, AC in directions indicated by the order of the letters. Find the magnitude of the resultant force correct to the tenth of a pound and the position of the points where its line of action cuts the sides AD, BC, correct to an inch.
Prove that two couples of equal moment and acting in the same plane are equivalent. \(AB\) is a rod, to whose ends are fixed small rings which slide, one on each, on smooth fixed horizontal rods \(OA, OB\) inclined at an acute angle \(\theta\): to a small ring which slides along \(AB\) is attached a string which passes over a smooth pulley at \(O\) and supports a weight \(W\). Prove that the couple which must be applied to \(AB\) in the plane of the rods to maintain equilibrium is \(W \cdot AB \cdot \sin(B-A)/\sin\theta\).
Two uniform rods \(AB\) and \(CD\) each of weight \(W\) and length \(a\) are smoothly jointed together at a point \(O\), where \(OB\) and \(OD\) are each of length \(b\). The rods rest in a vertical plane with the ends \(A\) and \(C\) on a smooth table and the ends \(B\) and \(D\) connected by a light string. Prove that the reaction at the joint is \(\dfrac{aW}{2b}\tan\alpha\), where \(\alpha\) is the inclination of either rod to the vertical.
Seven equal bars jointed together so as to form three triangles ABE, BED, BDC are placed in a vertical plane with ABC horizontal and ED above it. A weight \(W\) (so large that the weight of the bars may be neglected) is placed at E and the system is supported at A and C. Draw the stress diagram and indicate which bars are in tension.