Explain how the resultant of a three-dimensional system of forces may in some circumstances be a couple, and show that in this case the straight lines about which the system has zero moment must be parallel to the same plane. Prove that forces acting along and inversely proportional to the lengths of the perpendiculars from the vertices of any tetrahedron on to the opposite faces are in equilibrium.
A thin uniform rigid rod of weight \(W\) resting on a rough peg at \(A\) and supported from above by a similar peg at \(B\) is in equilibrium under its own weight and the reactions of the pegs. If the coefficient of friction is the same at both pegs, show that it cannot be less than \(\mu_0 = \frac{AB \tan\theta}{AC+BC}\), where \(C\) is the centre of the rod and \(\theta\) its inclination to the horizontal. An additional weight \(W'\) is attached to a certain point \(D\) of the rod such that equilibrium is still maintained but with increase of \(W'\) is eventually destroyed. Show that the rod will tend to turn or slip as the coefficient of friction is greater than or less than \(\tan\theta\).
A heavy uniform horizontal beam of length \(2l\) rests symmetrically on two supports which are at a distance \(2a\) apart on a horizontal level. Illustrate, by a simple sketch, the value of the bending moment for every point of the beam. If \(a=(4-2\sqrt{3})l\), find the points on the beam where the bending moment is zero.
A uniform heavy flexible chain hangs under gravity with its ends attached to light smooth rings which can slide on a horizontal rod. The rod is rotated about a vertical axis intersecting it, with constant angular velocity \(\omega\), and the chain is at relative rest in a vertical plane with the axis of rotation as an axis of symmetry. \(x\) is measured horizontally from the axis of rotation and \(y\) is measured vertically upwards from the lowest point of the chain. Prove that the tension at any point of the chain is given by \[ T=T_0 + pgy - \frac{1}{2}p\omega^2x^2, \] where \(T_0\) is the tension at the lowest point and \(\rho\) is the line density of the chain. Prove that the total length of the chain is \[ 2y_1 - \frac{\omega^2}{g}x_1^2 + \frac{2T_0}{\rho g}, \] where \(x_1, y_1\) are the values of \(x\) and \(y\) at one of the rings.
A small sphere is projected with velocity \(V\) in a vertical plane from a point \(O\) and subsequently strikes a plane through \(O\), with the line of greatest slope in the vertical plane and inclined at angle \(\alpha\) to the horizontal. Find the range \(x\) up the plane when \(\theta\) is the inclination of the velocity of projection to the line of greatest slope of the plane. If \(e\) is the coefficient of restitution of the impact of the sphere and the plane, prove that if \(2\tan\theta=(1-e)\cot\alpha\), the distances between successive impacts are in the decreasing ratio \(e^2\).
A bead of mass \(m\) moves on a smooth wire bent in the form of a circle of radius \(a\) which is held fixed in a vertical plane. An elastic string of natural length \(\mu a\) (\(0<\mu<2\)) has one end attached to the bead and the other end to the highest point of the wire. When the bead is released from rest at the position where the string is just taut, it is found that it comes to rest again at the lowest point of the wire. Prove that \(\mu\) must be the positive root of the equation \[ x^2+x(2+n)-2n=0, \] where \(n\) is the ratio of the modulus of the string to the weight of the bead.
A uniform chain of total mass \(m\) and length \(l\) is released from rest when held vertically with its lower end just touching the bottom of the interior of a bucket of mass \(M\). When half the chain has fallen into the bucket, the bucket itself is released and allowed to fall. It may be assumed that the impact of the chain and bucket is completely inelastic. In the subsequent motion, \(x\) is the distance the bucket has fallen and \(y\) is the length of chain remaining above the bucket. Show that the momentum is given by \[ (M+m)\dot{x} - my\dot{y}/l. \] Show further that the velocity of the chain at the same moment relative to the bucket is \[ (gl)^{\frac{1}{2}} \left(M+\frac{m}{2}\right) / \left(M+m\left(1-\frac{y}{l}\right)\right). \]
A compound pendulum has a detachable rider which it can shed as it passes through its equilibrium position in one sense and can resume as it passes through it in the reverse sense. The equilibrium position is the same with or without the rider attached. The moment of inertia about the horizontal axis from which the pendulum swings is \(I_1\) with the rider attached and \(I_2\) without it. The pendulum carrying the rider is released from rest at an angular displacement \(\alpha\) from the equilibrium position and first sheds and then resumes the rider as it passes and repasses the equilibrium position, and instantaneously comes to rest again on the original side at an angular displacement of \(\beta\). Prove that \[ I_1^2(1-\cos\beta) = I_2^2(1-\cos\alpha). \]
A smooth rigid wire bent in the form of a circle of radius \(a\) and centre \(C\) is constrained to rotate in its own plane (horizontal) with constant angular velocity \(\omega\) about a point \(A\) of its circumference. A bead \(P\) can move on the wire and \(\theta\) is the angle \(ACP\) measured from \(CA\) in the same sense as \(\omega\). By considering the acceleration of the bead along the tangent to the wire at \(P\), show that \[ \ddot{\theta} = \omega^2\sin\theta. \] If \(\dot{\theta}=2\omega\) when the line \(ACP\) is a diameter, prove that in the subsequent motion \[ \dot{\theta} = 2\omega \sin\frac{\theta}{2}. \]
A sphere of radius \(b\) resting on the top of a fixed rough hemisphere of radius \(a\) with horizontal base is displaced slightly and rolls down the hemisphere without slipping. Show that the sphere will leave the surface of the hemisphere when the line joining the centre of the sphere to that of the hemisphere is inclined to the upward vertical at angle \(\theta\), where \[ \cos\theta = \frac{2}{7}\left(3+\frac{\kappa^2}{b^2}\right), \] \(\kappa\) being the radius of gyration of the sphere about a diameter.