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.
Find the period of oscillation of a particle which moves in a straight line under the action of a force directed to a fixed point in the line and proportional to the distance from that point. Two equal particles connected by an elastic string which is just taut lie on a smooth table, the string being such that the weight of either particle would produce in it an extension \(a\). Prove that, if one particle is projected with velocity \(u\) directly away from the other, each will have travelled a distance \(u\pi\sqrt{(a/8g)}\) when the string first returns to its natural length.
Four forces acting along the sides of a quadrilateral are in equilibrium; prove that the quadrilateral is plane. If also the quadrilateral can be inscribed in a circle, prove that each force is proportional to the opposite side.
A uniform rod \(ACB\), of length \(2a\), is supported against a rough vertical wall by a light inextensible string \(OC\) attached to its middle point \(C\). The other end of the string is attached to a fixed point \(O\) on the wall. Shew that the rod can rest with \(C\) at any point of a circular arc, whose extremities are at perpendicular distances \(a\) and \(a\cos\lambda\) from the wall, where \(\lambda\) is the angle of friction.
A weight \(W\) rests upon a rough plane (\(\mu=\frac{1}{\sqrt{3}}\)) inclined at \(45^\circ\) to the horizontal, and is connected by a light string passing through a smooth fixed ring \(A\), at the top of the plane, with a weight \(\frac{W}{3}\) hanging vertically. The string \(AW\) makes an angle \(\theta\) with the line of greatest slope in the plane. Prove that the greatest possible value of \(\theta\) for equilibrium is \(\sin^{-1}\frac{1}{3}\).
An aeroplane has a speed of \(v\) miles per hour, and a range of action (out and home) of \(R\) miles in calm weather. Prove that in a north wind of \(w\) miles per hour, its range of action in a direction \(\theta\) east of north is \[ \frac{R(v^2-w^2)}{v(v^2-w^2\sin^2\theta)^{\frac{1}{2}}}. \] Find also the direction in which its range is a maximum.