A light inextensible chain passes round two toothed wheels, of radii \(a_1, a_2\) and moments of inertia \(I_1, I_2\) respectively, which are capable of rotation, without friction, about fixed parallel axes passing through the centres of the wheels; the chain lies in a plane perpendicular to these axes. A couple is applied to the wheel of radius \(a_2\); if the chain cannot withstand a tension greater than \(T_0\), find the greatest couple that can safely be applied. Find also the corresponding rate of increase of kinetic energy of the system. (The tension in the slack part of the chain can be ignored.)
Prove that any system of coplanar forces is equivalent to a force acting at a given point, together with a couple (where either the force or the couple, or both, may vanish). A system of coplanar forces, each of which has a fixed point of application, is in equilibrium. Prove that if the line of action of each force is rotated anticlockwise through the same angle \(\alpha\) (without change of magnitude) then the new system is equivalent to a couple of moment proportional to \(\sin\alpha\).
Five uniform rods \(AB, BC, CD, DE\) and \(EF\), each of length \(2a\) and weight \(W\) are freely jointed together at \(B, C, D\) and \(E\) to form a chain. The rods \(AB, EF\) can turn freely about fixed points \(A, F\) respectively, such that the line \(AF\) is horizontal and of length \(2(\sqrt{3}+1)a\). \(A\) is joined to \(C\), and \(D\) to \(F\), by strings each of length \(2\sqrt{3}a\). Find, by the method of virtual work, the tension in each string when the system hangs in equilibrium.
Two uniform rods \(AB, BC\), equal in weight and length, are freely jointed together at \(B\), and stand in a vertical plane with the ends \(A\) and \(C\) on a rough horizontal plane, each rod being inclined at an angle \(\theta\) to the horizontal. The coefficient of friction at \(A\) and \(C\) is \(\mu\). Show that, for equilibrium, \(\mu \ge \cot\theta\). A gradually increasing force is applied at \(B\) in a direction parallel to \(AC\). Determine how the way in which equilibrium is broken depends on the values of \(\mu\) and \(\theta\).
A particle of weight \(W\) is free to move on a smooth elliptical wire fixed with its major axis, of length \(2a\), vertical and its minor axis, of length \(2a/\sqrt{3}\), horizontal. The particle is attracted towards the highest point of the ellipse by a force which is of magnitude \(Wr/a\) when the particle is distant \(r\) from the highest point. Obtain an expression for the potential energy of the particle when its coordinates, referred to vertical and horizontal axes through the centre of the ellipse, the positive direction of the vertical axis being downwards, are \[ \left( a \cos\theta, \frac{a}{\sqrt{3}}\sin\theta \right). \] Hence, or otherwise, show that the values \(\theta=0\) and \(\theta=\pi/3\) give positions of equilibrium, and determine the stability of each of them.
A particle of mass \(m\) is attached by an inextensible string of length \(l\) to a ring, also of mass \(m\), which can slide on a fixed smooth horizontal rod. The system is released from rest with the string taut, nearly vertical, and in a vertical plane below the rod. Prove that the ring and the particle perform approximately simple harmonic oscillations, and find their period.
Four uniform bars \(AB, BC, CD\) and \(DA\) of length \(a\) and weights \(w, 2w, 2w\) and \(w\) respectively are freely jointed at \(A, B, C\) and \(D\). \(A\) and \(C\) are connected by a light elastic string of natural length \(a\). When the system is hung from \(A\), the length of \(AC\) is observed to be \(\frac{9}{8}a\). Show that the modulus of elasticity of the string is \(7w\). Find the reactions at the joints \(B\) and \(D\).
A uniform rod \(ABCD\) of length \(3l\) and weight \(W\) rests horizontally on a peg at \(B\), where \(AB=l\), and passes just under a peg at \(C\), where \(AC=2l\); a weight \(W'\) is attached to \(A\). Find the minimum value of \(W'\) that will maintain the rod in a horizontal position. If \(W'=W\), find the bending moment at all points of the rod and illustrate by a diagram. What is the maximum numerical value of the bending moment?
A uniform rod of weight \(W\) is placed with one end on a rough horizontal plane with the coefficient of friction \(\mu\) and the other end resting against a smooth vertical wall, such that the vertical plane containing the rod is perpendicular to the wall. Show that, for equilibrium to be possible, the inclination \(\theta\) of the rod to the horizontal must satisfy \[ \cot\theta \le 2\mu. \] If this condition is satisfied and also \(\cot\theta > \mu\), show that the maximum weight that can be attached to the highest point of the rod without breaking the equilibrium is \[ \frac{2\mu \tan\theta-1}{2(1-\mu\tan\theta)} W. \] What is the corresponding result if \(\cot\theta \le \mu\)?
A right circular cone of height \(h\) and volume \(\frac{1}{3}\pi a^2 h\) is made of non-uniform material whose density is proportional to the perpendicular distance from the base of the cone. Find the position of the centre of gravity. If the cone rests with its base on a perfectly rough horizontal plane with a small bead whose mass is equal to the mass of the cone attached to the vertex, show that the plane may be tilted through an angle \(\tan^{-1}(10a/7h)\) before the cone falls over.