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.
Two smooth spheres of equal mass whose centres are moving with equal speeds in the same plane, collide in such a way that at the moment of collision the line of centres makes an angle \(90^\circ-\beta\) with the direction bisecting the angle \(\alpha\) between the velocities before impact. Shew that after impact the velocities are inclined at an angle \(\tan^{-1}(\tan\alpha\cos 2\beta)\), the collision being perfectly elastic.
A heavy particle is attached by a light elastic string to a fixed point \(A\) on a rough plane whose inclination to the horizontal is \(\alpha\). Originally the string is unstretched, and lying along a line of greatest slope, \(A\) being the highest point of the string. Describe the motion in general terms, and shew that it will be all in one direction unless the coefficient of friction is less than \(\frac{1}{2}\tan\alpha\).
The propulsive horse-power required to drive a ship of mass 16,500 tons at a steady speed of 30 feet per second is 18,000. Assuming that the resistance is proportional to the square of the speed, and that the engines exert a constant propulsive force on the ship at all speeds, prove that the initial acceleration, when the ship starts from rest, is \(\frac{1}{5}\) feet per sec. per sec.; and that it attains a speed of 20 feet per sec. in \(\frac{5}{2}\log_e 5\) minutes.
If \[ y = ax\cos\left(\frac{n}{x}+b\right), \] prove that \[ x^4 \frac{d^2 y}{dx^2} + n^2 y=0. \] Prove that \(x^{1/x}\) is a maximum when \(x=e\).
If \[ y=a+x\log y, \] prove that when \(x\) is zero \[ \frac{dy}{dx} = \log a \quad \text{and} \quad \frac{d^2 y}{dx^2} = \frac{1}{a}(\log a)^2. \] If \(\lambda\) is a variable parameter, find the envelope of the family of straight lines \[ x(1+\lambda^2)+2\lambda y = a(1-\lambda^2). \]