Problems

The force of attraction between two particles of masses \(m, M\) is \(\gamma\frac{mM}{r^2}\), where \(\gamma\) is a constant and \(r\) is their distance apart. Two particles of masses \(m, 2m\) are at rest at a distance \(a\) apart when each receives an impulse \(\sqrt{\frac{2\gamma m^3}{a}}\) directly away from the other. Find the time that elapses before the particles next pass through their initial positions.

Discuss the absolute and gravitational units of force and the relations between them. Two pans each of mass \(m\) are connected by a light inextensible string passing over a smooth pulley and the pans hang freely in equilibrium. A uniform chain of length \(l\) and mass \(m\) is held over one pan with its lower end just touching it. If the chain is released from rest, find the time that elapses before it is all coiled up in the pan, neglecting the finite size of the coil produced.

Define the coefficient of restitution between two bodies. A smooth circular hoop lies on a smooth horizontal table to which it is rigidly fastened. A particle is projected from a point \(A\) of the hoop and after striking the hoop again at \(B\) and \(C\) it returns to \(A\). Prove that the ratio of the time taken by the particle to describe the perimeter of the triangle \(ABC\) to the time taken to describe the distance \(AB\) is \(\dfrac{2(1-e^3)}{e(1-e^2)}\), where \(e\) is the coefficient of restitution between the particle and the hoop.

Four equal particles \(A, B, C, D\) rest on a smooth horizontal plane at the vertices of a parallelogram in the order stated. \(C\) is joined to \(B\) and \(D\) by light inextensible strings which are taut, and the angle between these strings is \(\pi-\theta\), where \(\theta < \frac{\pi}{2}\). If \(A\) is projected towards \(B\), shew that the initial direction of motion of \(B\) is inclined to \(CB\) at an angle \[ \tan^{-1}\frac{(3+\sin^2\theta)\tan\theta}{1+\sin^2\theta}. \]

If \(O, A, B\) are three points in a vertical plane and if it is desired to project a particle from \(O\) to pass through \(A\) and \(B\) and to have the minimum speed of projection, prove that \(A, B\) and the focus of the trajectory must be collinear. Find the minimum speed of projection for the case in which \(BA\) is of length 5 feet perpendicular to \(AO\), and inclined to the horizontal at an angle \(\sin^{-1}\frac{3}{5}\).

A particle of mass \(m\) is free to move in a thin smooth uniform straight tube of mass \(3m\) and length \(a\). The tube can turn freely in a horizontal plane about one end which is fixed. Initially the tube has angular velocity \(\Omega\) and the particle is at rest relative to the tube at its mid-point. Find the velocity with which the particle leaves the tube.

Prove that, if \(bc+p^2 \neq 0\), then the equations \begin{align*} ax + qy - rz &= a, \\ -qx + by + pz &= 0, \\ rx - py + cz &= 0 \end{align*} have an infinite number of solutions if, and only if, \[ a=0 \quad \text{and} \quad abc + ap^2 + br^2 + cq^2 = 0. \]

The roots of the equation \[ x^3+px^2+qx+r=0 \] are \(\alpha, \beta, \gamma\). Shew that \(\alpha^3, \beta^3, \gamma^3\) are the roots of the equation \[ \begin{vmatrix} p & q & x+r \\ q & x+r & px \\ x+r & px & qx \end{vmatrix} = 0. \]

Shew that the geometric mean of \(n\) positive numbers is not greater than their arithmetic mean. If \(x, y, z\) are positive numbers such that \(x+y+z=1\), find the greatest value of \(x^ay^bz^c\).

If \(0 < x < 1\), shew that \(n^2x^n \to 0\), as \(n \to \infty\). Find the limit as \(n \to \infty\) of \[ \frac{x^{2n}}{n+x^{2n}}; \] distinguish between the two cases.