The figure represents an inextensible string attached to a fixed point \(O\), passing under a rough pulley \(B\) which hangs in the loop, passing over another rough pulley \(A\) with fixed centre and supporting a mass \(M\) at its other end. All the strings are vertical. \(M_1\) is the combined mass of the pulley \(B\) and the attached load. \(I, I_1\) are the moments of inertia of the pulleys about their centres, and \(a,b\) their radii. Find the acceleration with which \(M\) moves. % Figure shows a pulley system. A fixed point O at the top. % String goes from O, under a movable pulley B, over a fixed pulley A. % Mass M hangs from the end of the string after pulley A. % Pulley B has a mass M1 attached to it.
Prove that a system of forces acting in one plane on a rigid body can be reduced to a force in a given line \(\lambda\) and a force through a given point \(O\) (not lying on \(\lambda\)). Prove also that the reduction is unique. Forces \(P, 2P, P, 2P, P, 2P\) act in the sense indicated along the sides \(AB, BC, CD, DE, EF, FA\) of a regular hexagon \(ABCDEF\). Reduce the system to a force in \(AB\) and a force through \(C\).
A uniform rod \(AB\), of length \(2a\) and weight \(W\), is freely hinged at \(B\) to a uniform rod \(BC\), of length \(2a\) and weight \(3W\). The ends \(A, C\) are freely attached to light rings, which slide on a rough horizontal rail. Find the horizontal and vertical components of the reaction of the rail on the ring at \(A\) when the rods hang at rest with each rod inclined at an angle \(\theta\) to the vertical. What is the greatest possible distance between the rings if the coefficient of friction between either ring and the rail is \(1/2\)?
A non-uniform elastic string is such that the modulus of elasticity at a point of the string varies uniformly, in the unstretched state, from \(\lambda_1\) at one end to \(\lambda_2\) at the other. Prove that the modulus of elasticity for the string as a whole (i.e. the tension needed to double its length) is \((\lambda_2-\lambda_1)/\log(\lambda_2/\lambda_1)\).
A uniform heavy inelastic string, whose weight per unit length is \(w\), hangs freely under gravity with its ends held at the same level. Prove that, if \(\psi\) is the inclination to the horizontal of the tangent, and \(T\) is the tension, at a point of the string whose distance along the string from the lowest point is \(s\), then \begin{align*} s &= c \tan\psi, \\ T &= wc \sec\psi, \end{align*} where \(c\) is a constant. Two uniform rods \(AB\) and \(CD\), each of length \(2a\) and weight \(2aw\), are freely attached at \(A\) and \(C\) to fixed points at the same level. The ends \(B\) and \(D\) are joined by a uniform string, of length \(2a\) and weight \(2aw\), and the system hangs in equilibrium. If \(\theta\) is the inclination of either rod to the vertical, and \(\beta\) is the inclination to the horizontal of the tangent to the string at \(D\), prove that \[ 2\tan\theta\tan\beta=1. \]
A particle of mass \(m\) moves on a straight line under a force \(mn^2r\) towards a fixed point \(O\) of the line, where \(r\) denotes distance from \(O\). Prove that the motion is periodic, with period \(2\pi/n\). A particle of mass \(m\) is attached to the mid-point of a light uniform elastic string, of natural length \(2a\). When the ends of the string are attached to fixed points at the same level and at a distance \(2a\) apart, and the particle hangs in equilibrium, the stretched length of the string is \(2b\). Prove that the period of a small vertical oscillation about the position of equilibrium is \(2\pi/n\), where \[ n^2 = \frac{b^2+ba+a^2}{b^2\sqrt{(b^2-a^2)}}g. \]
The engine of a car of mass \(m\), travelling on a level road, works at a constant rate \(R\), and the resistance to motion is proportional to the speed. If the steady speed at which the car can travel is \(w\), shew that the car, starting from rest, acquires a velocity \(v\) in a distance \[ \frac{mw^3}{2R}\left(\log\frac{w+v}{w-v} - \frac{2v}{w}\right). \] If the mass of the car is 18 cwt., the rate of working is 10 horse-power, and the steady speed is 50 miles per hour, find in feet the distance in which the car acquires a speed of 30 miles per hour.
A particle moves under gravity, being projected from a point \(O\) with velocity \(\sqrt{(2gh)}\). Prove that the path of the particle is a parabola whose directrix is a horizontal line at height \(h\) above \(O\). A ball is thrown over a wall. The ball is projected from a point \(O\) in the vertical plane through \(O\) perpendicular to the wall, and the velocity of projection is \(\sqrt{(2gh)}\). The top of the wall is at a distance \(r\) from \(O\), and at a vertical height \(q\) above \(O\). Prove that \(q+r < 2h\), and that the direction of projection lies within an angle \(\theta\), where \[ \cos\theta = \frac{r^2+2hq-q^2}{2hr}. \]
Find the formulae for the radial and transverse components of acceleration of a particle moving in a plane, the position of the particle at time \(t\) being described by the polar coordinates \(r, \theta\). A particle \(P\) of mass \(m\) moves in a plane under the action of a centre of force \(O\) (i.e. a force in the line \(OP\)), and of a force of magnitude \(2mkv\) at right angles to the direction of motion, where \(v\) is the speed and \(k\) a constant. The particle is projected from \(O\) with velocity \(v_0\). Prove that the angular velocity \(\dot\theta\) remains constant throughout the motion. Find the path of the particle (i) when the central force is an attraction \(3mk^2r\) towards \(O\), (ii) when the central force is a repulsion \(mk^2r\) away from \(O\).
A rigid body is free to swing, as a pendulum, about a horizontal axis. Find the length of the equivalent simple pendulum. A uniform rod \(AB\), of mass \(M\) and length \(2a\), hangs freely from a fixed pivot at \(A\), and a particle of mass \(6M\) is attached to the rod at distance \(x\) from \(A\). Prove that the length of the equivalent simple pendulum is \[ \frac{2}{3} \left( \frac{9x^2+2a^2}{6x+a} \right). \] If \(x\) can take any value from \(0\) to \(2a\), shew that the least value of this length is \(2a/3\), and find the greatest value.