Explain what is meant by a couple and define its moment. From the definition, show that two couples of equal moment are equivalent if they act in the same plane or in parallel planes. A system of forces in a plane is such that the total moment of the forces about any point of the plane can be ascertained. Show that from a knowledge of moments about three suitably chosen points can be determined the nature and magnitude of the resultant of the system, whether it reduces to a couple or a force.
A slightly flexible heavy uniform beam of length \(2a\) rests with its two ends at the same horizontal level. When the ends are simply supported without being gripped, the middle of the beam sags to a depth \(h\) below its ends. Show that if the rod is supported with the ends gripped at a slight elevation \(\alpha\) to the horizontal so that the midpoint is at the same level as the ends, the value of \(\alpha\) is given approximately by \(5ax=2h\).
A uniform flexible chain of line density \(w\) is held at rest under gravity in contact with a smooth vertical curve in the shape of a catenary whose vertex is at a height \(c\) above its directrix which is horizontal. The chain is in contact with the curve at all points of its length and the tension at its lowest point is \(T_0\). Prove that the pressure \(R\) per unit length of arc at any point on the curve due to the chain is given by \(R = \left(\frac{T_0}{c}-w\right)\cos^2\psi\), where \(\psi\) is the inclination to the horizontal of the tangent to the curve.
A non-uniform rigid rod has its ends attached to light rings which can slide on a rigid rough wire in the shape of a circle held in a vertical plane. The coefficient of friction at both ends is \(\mu\) and is less than unity, and the rod subtends a right angle at the centre of the circle. Find the range of values for the angle of inclination to the horizontal of the rod in equilibrium for varying positions of its centre of mass, and verify that the range of values varies between \(2\lambda\) and \(4\lambda\), where \(\tan\lambda=\mu\).
Derive the usual formulae for the tangential and normal accelerations of a particle moving in a plane curve. A particle moves under gravity in a medium offering a constant resistance to its motion equal to \(n\) times its weight. Prove that if at any instant the resultant velocity is \(w\), with horizontal and vertical components \(u\) and \(v\) respectively, then \((w+v)^n \cdot u^{-(n+1)}\) is constant.
A particle \(P\) moves with acceleration \(\lambda r^{-3}\) directed towards a fixed origin \(O\), where \(r\) is the length of \(OP\) and \(\lambda\) is a positive constant. Using polar coordinates, when \(r=d\), and \(\theta=0\) the direction of the velocity \(v\) is inclined at \(\dfrac{\pi}{4}\) to the outward radius vector, and \(v^2\) is positive and greater than unity and denoted by \(n^2\). Establish the equation of motion in the form \(a^2 (\frac{du}{d\theta})^2 + u^2 = n^2u\), where \(u=1/r\), and prove that the greatest value of \(r\) during the motion is given by \(an(n^2-1)^{-\frac{1}{2}}\).
A heavy particle \(P\) can move under gravity in a vertical straight line \(AB\) and is attached to the ends of two similar elastic strings \(PA\) and \(PB\) of natural length \(l\). The particle can rest in equilibrium at a point \(O\) between \(A\) and \(B\), the lower string \(PB\) being just in tension while the upper string \(AP\) is extended a distance \(a\). If the particle is released from rest when at a height \(h\) above \(O\), where \(h>a\), find the appropriate equations of motion for the downward path, and prove that the particle first comes to rest when at a depth \(x\) below \(O\) given by \(x^2 = h^2+2ah-a^2\).
A smooth uniform sphere of mass \(M\) collides with a similar stationary sphere of mass \(m\). The coefficient of restitution is \(e\) and at impact the velocity of \(M\) is inclined at an angle \(\alpha\) to the line of centres. Given that \(M>em\), find the angle between the velocities of the spheres after impact, and show that the direction of the velocity of the sphere \(M\) is deflected through an angle \(\theta\), where \(\tan\theta[(M+m)\tan^2\alpha+M-em] = m(1+e)\tan\alpha\).
A rigid rod whose centre of mass is at its midpoint is moving in a plane when it strikes an inelastic peg at a point \(X\) distant \(x\) from its centre. If immediately before impact the velocity \(u\) of the midpoint is perpendicular to the rod and the angular velocity is \(\omega\), find the angular velocity immediately after impact, and show that the loss of kinetic energy is \(\frac{1}{2}\dfrac{M k^2}{k^2+x^2}(u+\omega x)^2\), where \(M\) is the mass and \(Mk^2\) the moment of inertia about the centre.
Prove that the period of small oscillations of a uniform hemisphere in rocking motion with its curved surface in contact with a perfectly rough horizontal plane is given by \(2\pi\sqrt{\frac{26a}{15g}}\), where \(a\) is the radius of the hemisphere.