Given the motion of two smooth spheres before impact, write down equations to determine their motion after direct impact. A smooth inclined plane of slope \(\alpha\) and mass \(M\) is free to move on a smooth horizontal plane in a direction perpendicular to its edge. A spherical ball of mass \(m\) is dropped on it. Prove that the ball will rebound in a direction inclined to the horizontal at an angle \[ \tan^{-1}\left\{\frac{(M+m)\sin^2\alpha - Me\cos^2\alpha}{M(1+e)\sin\alpha\cos\alpha}\right\}, \] where \(e\) is the coefficient of restitution.
Reduce a system of given coplanar forces to a force or a couple. A, B, C, D are successive corners of a square whose side is 100 inches in length. Forces 2, 3, 1, 5, 4 lbs. act respectively along AB, BC, DC, DA, AC in directions indicated by the order of the letters. Find the magnitude of the resultant force correct to the tenth of a pound and the position of the points where its line of action cuts the sides AD, BC, correct to an inch.
Prove that two couples of equal moment and acting in the same plane are equivalent. \(AB\) is a rod, to whose ends are fixed small rings which slide, one on each, on smooth fixed horizontal rods \(OA, OB\) inclined at an acute angle \(\theta\): to a small ring which slides along \(AB\) is attached a string which passes over a smooth pulley at \(O\) and supports a weight \(W\). Prove that the couple which must be applied to \(AB\) in the plane of the rods to maintain equilibrium is \(W \cdot AB \cdot \sin(B-A)/\sin\theta\).
Two uniform rods \(AB\) and \(CD\) each of weight \(W\) and length \(a\) are smoothly jointed together at a point \(O\), where \(OB\) and \(OD\) are each of length \(b\). The rods rest in a vertical plane with the ends \(A\) and \(C\) on a smooth table and the ends \(B\) and \(D\) connected by a light string. Prove that the reaction at the joint is \(\dfrac{aW}{2b}\tan\alpha\), where \(\alpha\) is the inclination of either rod to the vertical.
Seven equal bars jointed together so as to form three triangles ABE, BED, BDC are placed in a vertical plane with ABC horizontal and ED above it. A weight \(W\) (so large that the weight of the bars may be neglected) is placed at E and the system is supported at A and C. Draw the stress diagram and indicate which bars are in tension.
A cylinder rests in equilibrium on a table. Shew that if the radius of curvature of any cross-section at its point of contact is greater than the height of the centre of gravity, the equilibrium is stable. Shew that the stable equilibrium of an elliptic cylinder lying with its generators horizontal on a table cannot be rendered unstable by loading it at the top if \(e > 1/\sqrt{2}\).
Shew that if a perfectly elastic sphere collides with another at rest, and their lines of motion after impact are at right angles, their masses must be equal.
A body moves in a straight line under the action of a force acting along that line. If a curve be drawn, with the distance described as abscissa and the corresponding force as ordinate, shew that an area on this diagram represents the work done by the force. A motor car is travelling along a level road with a constant velocity of \(V\) feet per second, the resistances to motion being equivalent to a constant back pull of \(a\) lbs. weight. The car then comes to a hill where the resistance to motion (including gravity) is \(b\) lbs. weight, and after the velocity has again become constant the engine works at the same constant power as on the level. If, while the velocity is varying, the tractive pull alters uniformly with the distance from its first constant value to its next constant value, shew that the distance travelled along the hill before the velocity becomes constant is \(\dfrac{a+b}{b^2-a^2} \dfrac{MV^2}{g}\) feet, where \(M\) is the mass of the car in pounds.
Shew that the path of a projectile in vacuo under gravity is a parabola, and express the velocity at any point in terms of the depth of the point below the directrix. A ball is projected with velocity \(V\) from a point on the ground at distance \(a\) from a wall of height \(b\). Prove that the least velocity for it to clear the wall is given by \(V^2 = g\{b+\sqrt{(a^2+b^2)}\}\).
Determine the acceleration of a point describing a circle with uniform speed. A small ring fits loosely on a rough spoke (length \(a\)) of a wheel which can turn about a horizontal axle and the ring is originally at rest in contact with the lowest point of the rim: if the wheel is now made to revolve with uniform angular velocity \(\omega\), prove that the angle \(\theta\) through which the wheel will turn before the ring slides is given by the equation \[ g\cos(\theta-\lambda)+a\omega^2\cos\lambda=0, \] where \(\lambda\) is the angle of friction.