A particle of mass \(m\) is attached by a string to a point on the circumference of a fixed circular cylinder of radius \(a\) whose axis is vertical, the string being initially horizontal and tangential to the cylinder. The particle is projected with velocity \(v\) at right angles to the string along a smooth horizontal plane so that the string winds itself round the cylinder. \par Shew (i) that the velocity of the particle is constant, \par (ii) that the tension in the string is inversely proportional to the length which remains straight at any moment, \par (iii) that if the initial length of the string is \(l\) and the greatest tension the string can bear is \(T\), the string will break when it has turned through an angle \(l/a-mv^2/aT\).
Shew that if a number of particles connected by inelastic strings move under no forces, their linear momentum and energy are constant. \par Three equal particles \(A, B, C\) connected by inelastic strings \(AB, BC\) of length \(a\) lie at rest with the strings in a straight line on a smooth horizontal table. \(B\) is projected with velocity \(V\) at right angles to \(AB\). Shew that the particles \(A\) and \(C\) afterwards collide with relative velocity \(\frac{2V}{\sqrt{3}}\). \par If the coefficient of restitution is \(e\), find the velocities of the three particles when the string is again straight.
Shew how to find graphically the resultant of any number of given coplanar forces. \par A uniform plank of weight 30 lbs. and length 12 feet is supported at each end and carries two weights of 10 lbs. each at distances 4 and 5 feet from one end. Find graphically the pressures on the supports.
Two particles of a system of masses \(m_1, m_2\) are at \(A, B\). If these two particles are interchanged, prove that the centre of gravity of the whole system moves through a distance \(\frac{m_1-m_2}{\Sigma m} AB\) parallel to \(AB\).
Two light rods are freely jointed together at one end and the other ends carry weights \(W, W'\). The rods are in a vertical plane, each being supported by one of two smooth pegs on the same level. If there is equilibrium when the rods are at right angles, prove that \[ a^2W^2+b^2W'^2=c^2(W+W')^2, \] where \(a, b\) are the lengths of the rods and \(c\) the distance between the pegs.
Two cylinders of equal radius but different weights \(W, W' (W'>W)\) rest inside another cylinder which is fixed. All the cylinders are equally rough, their axes are horizontal and the line of contact of the two smaller cylinders is vertically below the axis of the fixed cylinder. Shew that equilibrium is impossible unless the coefficient of friction is \(> \frac{W'-W}{W'+W}\frac{\cos\alpha}{1+\sin\alpha}\) when \(2\alpha\) is the angle between the planes through the axis of the fixed cylinder and the axes of the smaller cylinders. Shew also that if the coefficient of friction has this value each of the two smaller cylinders will be on the point of rolling on the fixed cylinder.
\(ABCD\) is a rhombus formed of freely jointed light rods. \(AC\) is vertical, \(A\) being the higher end, and \(B, D\) are tied by strings of equal length to a fixed point in the line \(AC\). Weights \(W, W' (W>W')\) are suspended from \(A, C\). If the rhombus is constrained to remain in a vertical plane, prove that in the position of equilibrium the fixed point divides \(AC\) in the ratio \(W':W\).
An engine of 250 horse-power pulls a load of 150 tons up an incline of 1 in 75. Taking the road resistance to be 16 lbs. per ton, find the greatest speed attainable on the incline.
An elastic sphere strikes obliquely an equal sphere at rest. Find the angle through which the direction of motion is deflected, and prove that if the line of motion of the centre of the moving sphere before contact is tangential to the fixed sphere the angle of deflection is \(\tan^{-1}\frac{(1+e)\sqrt{3}}{5-3e}\), where \(e\) is the coefficient of restitution.
A particle is projected along the inner side of a smooth circle of radius \(a\), the velocity at the lowest point being \(u\). Shew that if \(u^2<5ga\) the particle will leave the circle before arriving at the highest point and will describe a parabola whose latus-rectum is \(2(u^2-2ga)^3/27g^2a^2\).