Establish the equivalence of the two definitions of simple harmonic motion, (i) as motion of a point in a straight line under a restoring acceleration proportional to the distance from a fixed point in that line, (ii) as the projection of uniform circular motion on a diameter of the circle. A mass \(M\) hanging at rest from a light spring stretches it by a length \(c\). Show that, if \(M\) is slightly displaced vertically and allowed to oscillate, the period of oscillation is that of the small vibrations of a simple pendulum of length \(c\). While the mass \(M\) is hanging at rest a second mass \(M\) moving vertically upwards with velocity \((2gc)^{\frac{1}{2}}\) strikes it, and the masses adhere. Show that in the subsequent motion the spring is always extended from its unstretched length. Find the greatest and the least extensions of the spring during the motion.
A heavy non-uniform flexible chain hangs in equilibrium between two fixed points in such a way that its weight per unit length at any point is proportional to the tension at that point. Show that, referred to certain rectangular axes \(Ox, Oy\), and with a suitable choice of the unit of length, the equation of the curve in which the chain hangs may be put in the form \(y = \log \sec x\).
A heavy uniform equilateral triangular plate \(ABC\) is fitted with three light studs at the vertices \(A, B, C\) and rests in a horizontal position with the studs in contact with the rough surface of a table. If a gradually increasing horizontal force is applied at the vertex \(A\) in a direction parallel to \(BC\), show that eventually the plate will begin to turn about a point on the circumcircle of the triangle \(ABC\). If \(W\) is the weight of the plate and \(\mu\) the coefficient of friction between the studs and the table, find the limiting value of the force that has to be applied.
A rectangular trapdoor of weight \(W\) can turn freely about smooth hinges attached at one edge which makes an angle \(\alpha\) with the horizontal. The door is held in equilibrium with its plane making an angle \(\beta\) with the vertical plane \(p\) through the line of hinges by means of a force \(F\) applied at the centre of gravity of the door in a direction perpendicular to \(p\). Find the size of the requisite force \(F\).
Show that the resultant of two forces represented by vectors \(\lambda \vec{OA}\) and \(\mu \vec{OB}\) is \((\lambda+\mu)\vec{OG}\), where \(G\) is the centroid of masses \(\lambda\) at \(A\) and \(\mu\) at \(B\). Generalise this result to find the resultant of forces represented by \(\lambda \vec{OA}, \mu \vec{OB}, \nu \vec{OC}, \dots\). A triangle is formed of three heavy uniform bars of lengths \(2a, 2b, 2c\), and weights \(w_a, w_b, w_c\), respectively. It is suspended from a fixed point by three strings of lengths \(p, q, r\) attached to the midpoints of the three bars, respectively. Show that in equilibrium the tensions in the strings are in the ratios \(ap:bq:cr\).
A rider in open flat country can move with speed \(v\), and he wishes to signal to a train travelling on a straight track with speed \(V\) (\(>v\)) too great to allow him actually to intercept the train. If \(R\) is the initial position of the rider, \(T\) the position of the train at the same instant, \(N\) the foot of the perpendicular from \(R\) to the track, with \(N\) in front of \(T\), and if \(s\) is the maximum signalling range, show that the rider can get within signalling distance if \[ s > RN(1-v^2/V^2)^{\frac{1}{2}} - TN.v/V. \]
A particle can be projected under gravity (\(g\)) with fixed speed \(U\) from a point \(O\) of a plane inclined to the horizontal at an angle \(\alpha\). Show that the region within range is bounded by an ellipse of eccentricity \(\sin\alpha\) having \(O\) as a focus, and find its area.
A particle of mass \(m\) is projected with velocity \(u\) along the central line of greatest slope of a smooth wedge of inclination \(\alpha\) and mass \(M\). The wedge is initially at rest on a smooth horizontal plane but is free to slide on it. Show that if \(u\) is upwards along the wedge, the height \(h\) that the particle rises above its initial level is \[ u^2(M+m\sin^2\alpha)/2g(M+m). \] Find also the distance that the wedge has moved when the particle has returned to the point on the wedge from which it started.
A light inextensible rope is fastened at one end to a fixed point \(O\), and passes first under a smooth light pulley carrying a load \(2M\) suspended from its centre, then over a smooth fixed light pulley, and finally carries a bucket of mass \(M\) at its other end. The system is in equilibrium with the parts of the rope not in contact with the pulleys vertical when an elastic ball of mass \(m\) drops vertically with velocity \(v\) on to the bottom of the bucket and rebounds vertically. Show that the velocity thereby given to the bucket is \[ 2mv(1+e)/(2m+3M), \] where \(e\) is the coefficient of restitution between the ball and the bucket. Find also the impulse on the ball.
A light elastic string of modulus \(\lambda\) and natural length \(a\) is fixed at one end and carries a particle of mass \(m\) at the other. When the particle is held at depth \(d\) (\(>a\)) below the fixed end of the string and released, it is found that in the subsequent motion it just rises to the fixed end. Show that \(\lambda(d-a)^2 = 2mgad\). Find also an expression for the time taken by the particle from its release to reach the fixed end of the string.