A sphere collides simultaneously with two other spheres which are at rest and in contact; all three spheres are smooth and equal to one another. Assuming that for each point of impact the relative velocity along the normal after the impulse is \(-e\) times the relative velocity along the normal before the impulse, shew that \[ \tan\theta' = \frac{2-e}{2-3e}\tan\theta, \] where \(\theta, \theta'\) are the angles that the velocity of the first sphere makes with the perpendicular to the line of centres of the other two spheres before and after contact.
State the conditions under which a body will remain in equilibrium when acted on by three non-parallel forces. A uniform pole \(AB\), 13 ft. long, is suspended by ropes \(CA\) and \(DB\) from two points \(C\) and \(D\) which are at the same level and less than 12 ft. apart, and hangs so that \(B\) is 4 ft. and \(A\) 9 ft. below the line \(CD\). If the rope \(DB\) is 5 ft. long, find the length of the rope \(CA\), and the ratio of the tensions in the two ropes.
A particle of mass \(m\) rests on a plane inclined at an angle \(\alpha\) to the horizontal, and the angle of friction between the particle and the plane is \(\phi\). A horizontal force \(P\), making an angle \(\beta\) with vertical planes containing lines of greatest slope, is just sufficient to move the particle horizontally. Prove that \[ \tan\beta = \frac{\tan\phi}{\sin\alpha} \quad \text{and} \quad P = \frac{mg\sqrt{\sin^2\alpha+\tan^2\phi}}{\cos\alpha}. \]
The cantilever frame shown in Fig. 1 is built up of light rods and freely hinged throughout. Find the thrusts or tensions in the rods due to a load of 5 tons at \(A\).
A light flexible belt passes over a fixed pulley and is in contact with it for an angle \(\theta\) of the pulley circumference. The tensions on the free ends of the belt are \(T_1\) and \(T_2\) and the coefficient of friction between belt and pulley is \(\mu\). By considering the equilibrium of an element of belt embracing an angle \(\delta\theta\) of the pulley circumference, prove that the belt will be on the point of slipping if \(T_1/T_2 = e^{\mu\theta}\).
The surface bounded by the parabola \(x^2=4ay\), the axis of \(y\) and the line joining the points \((0,h)\) and \((2\sqrt{ah}, h)\) is revolved about the axis of \(y\) to form a solid of revolution. Prove that a body of this shape, made of uniform material, will stand in stable equilibrium on a horizontal surface with its axis vertical if \(h \le 2a(1+\sqrt{2})\). Find the angle to which the axis will be tilted if \(h=5a\) and the body rests in equilibrium with the parabolic surface in contact with the horizontal plane.
A flywheel of mass 80 lb. is suspended with its axis vertical by three vertical cords placed equidistant from the axis and at the corners of an equilateral triangle of side 1 ft., and each rope is 5 ft. long.
A car has mass \(M\) and is subjected to a constant net propulsive force \(P\), while wind effects produce a force \(aV^2\), \(V\) being the velocity of the car relative to the air, and \(a\) being a constant. The car starts from rest with a following wind of velocity \(U\). Find the time taken in accelerating to a speed equal to the maximum obtainable in still air.
A particle hangs from a light inextensible string of length \(r\) attached at its upper end to a point on a vertical wall. It is projected with velocity \(2\sqrt{rg}\) perpendicular to the wall. Find the point at which the particle will subsequently hit the wall.
A uniform disc of radius \(r\) and mass \(M\) is freely pivoted at a point on its circumference and hangs in a vertical plane. It is struck by a horizontal impulse \(F\) which acts in the plane of the disc, and is distant \(h\) below the pivot. Find the impulsive force at the pivot, and prove that if \(h=r\) the energy imparted to the disc is \(F^2/3M\).