Three light strings are attached at points \(A\), \(B\), \(C\) to a circular hoop which is in a vertical plane where \(A\), \(B\), \(C\) are the vertices of an equilateral triangle, and the other ends of the strings support a particle of weight \(W\) at the centre of the hoop. If for any position of the hoop the tensions of the three strings are \(P, Q, R\), prove that \[ P^2+Q^2+R^2-QR-RP-PQ=W^2. \]
If known weights are attached to points on a light string, the ends of which are fixed, and if the directions of two portions of the string are known, shew how to find the direction of each portion of the string. \par Weights \(W, w, W\) are attached to points \(B, C, D\) respectively on a light string \(AE\), where \(B, C, D\) divide the string into four equal lengths. If the string can hang in the form of four sides of a regular octagon with the ends \(A\) and \(E\) attached to points on the same level, prove that \(W=(\sqrt{2}+1)w\).
Prove that a system of forces in a plane can be replaced by two forces in the plane, one acting along a given line and one acting through a given point. \par A rigid roof truss \(ABC\) is in the form of an isosceles triangle right-angled at \(B\) and rests on two walls at \(A\) and \(C\). It carries a weight \(W\) symmetrically distributed on the two sides, and due to wind pressure there is a force \(w\) uniformly distributed along \(BC\) perpendicular to \(BC\). If the reaction at \(C\) is vertical, find the horizontal and vertical components of the reaction at \(A\).
Establish the principle of virtual work for a lamina under the action of forces in its plane. \par Three similar uniform rods \(AB, BC, CD\) each of length \(a\) are smoothly jointed together at \(B\) and \(C\) and the ends \(A\) and \(D\) are smoothly jointed at points on the same level at a distance \(2a\) apart. A weight \(W\) hangs from a point of trisection of \(BC\) and the system is kept in equilibrium with \(BC\) horizontal by a force \(P\) acting along \(BC\). Prove that \(3\sqrt{3}P=W\), independently of the weight of the rods.
Two cylinders of unequal radii are placed with their axes parallel on a horizontal plane and a plank is laid across them with its length at right angles to the axes of the cylinders and its centre of gravity half-way between the points of contact. One cylinder is held fixed and the plane is sufficiently rough to prevent the other cylinder from slipping. Prove that if \(\mu\) is the coefficient of friction between the plank and either cylinder and if \(\alpha\) is the inclination of the plank to the horizontal, the system is in equilibrium if \(\mu > \frac{1}{2}(2\tan\alpha - \tan\alpha/2)\), and that the first cylinder can be held fixed by a couple of magnitude \(Wa\sin\alpha\), where \(W\) is the weight of the plank and \(a\) is the radius of that cylinder.
If \(A\) and \(B\) are points on a rod which is moving in any way in a plane, and if \(Oa\) and \(Ob\) represent the velocities of \(A\) and \(B\) at any instant, prove that \(ab\) is perpendicular to \(AB\). If \(C\) is any other point on the rod and if \(c\) divides \(ab\) in the same ratio as that in which \(C\) divides \(AB\), prove that \(Oc\) represents the velocity of \(C\) at the same instant. \par \(PQ, QR, RS\) are three rods in a plane jointed together at \(Q\) and \(R\), and with the ends \(P\) and \(S\) jointed to fixed supports. If a triangle \(Oqr\) is drawn with \(Oq, qr, ro\) perpendicular to \(PQ, QR, RS\) respectively for any position of the rods, prove that as the rods move through this position \(Oq\) and \(Or\) represent on the same scale the velocities of \(Q\) and \(R\).
A train starting from rest is uniformly accelerated until its velocity is 30 feet per second and then uniformly accelerated at half the previous rate until its velocity is 60 feet per second and after that the velocity remains uniform. The train takes 118 seconds to travel the first mile. Find the initial acceleration. \par If the train weighs 200 tons and the resistances to motion are equivalent to a back pull of 16 lbs. wt. per ton find the average horse-power and also the maximum horse-power at which the engine was working during the time it took to travel the first mile.
Prove that the path of a projectile under no forces but gravity is a parabola. \par An aeroplane is flying with constant velocity \(v\) and at constant height \(h\). Show that, if a gun is fired point blank at the aeroplane after it has passed directly over the gun and when its angle of elevation as seen from the gun is \(\alpha\), the shell will hit the aeroplane provided \(2(V\cos\alpha-v)v\tan^2\alpha = gh\), where \(V\) is the initial velocity of the shot, the path being assumed to be parabolic.
A particle of mass \(m\) moving with velocity \(u\) impinges on a particle of mass \(M\). If after the impact the component velocities of the mass \(m\) are \(u'\) and \(v'\) in directions along and perpendicular to the original direction of motion, find the component velocities of the mass \(M\) in these directions. \par If there is no loss of energy in the impact prove that the greatest value that \(v'\) can have is \(\frac{M}{M+m}u\), and that this occurs when the line of the impulse makes an angle of \(\frac{\pi}{4}\) with the original direction of motion.
A particle when hanging in equilibrium at the end of a light elastic string stretches it a distance \(a\). Prove that the period of vibration of the particle in a vertical line through its equilibrium position is the same as that of a simple pendulum of length \(a\). \par A light endless elastic string of unstretched length \(2b\) passes over two small smooth pegs on the same level distant \(b\) apart. A particle is attached to a point on the string and when the particle is in equilibrium the string forms the three sides of an equilateral triangle. Prove that the period of vibration of the particle in a vertical line is the same as that of a pendulum of length \(\frac{2\sqrt{3}}{7}b\).