Problems

Two equal particles are joined by a light inextensible string of length \(\pi a/2\) and rest symmetrically on the surface of a smooth circular cylinder of radius \(a\), the axis being horizontal. If the particles are slightly disturbed, show that one of the particles will leave the surface at a height \[ \frac{1}{5}(2\sqrt{2}-\sqrt{3})a \] above the axis of the cylinder. The motion takes place in one plane.

Prove that conics through four fixed points on a circle with centre \(O\) have their axes parallel to fixed directions, and that the locus of the centres of such conics is a rectangular hyperbola \(H\) whose asymptotes are parallel to the axes of the conics. \par Shew further that \(H\) meets each conic \(S\) of the pencil in the feet of the four normals from \(O\) to \(S\).

Evaluate \[ \int \frac{dx}{\tan x + c}. \] Shew that \[ \int_0^\pi \frac{(x-1)^4}{(x+1)^5} dx = \frac{1}{5} \quad \text{and} \quad \int_0^{\pi/2} \sqrt{\tan x} dx = \frac{\pi}{\sqrt{2}}. \]

Two particles of mass \(m\) and \(M\) are connected by a light inextensible string of length \(2l\) which passes through a smooth hole in a smooth horizontal table on which the mass \(M\) moves while \(m\) hangs vertically. Initially \(M\) is at rest, at a distance \(l\) from the hole, and it is then projected with velocity \(V\) at right angles to the string. Shew that if \(3MV^2 > 8mgl\), \(m\) will reach the hole with velocity \[ \sqrt{\frac{3MV^2-8mgl}{4(M+m)}}. \]

Two particles of mass \(m\) and \(2m\) are hanging in equilibrium attached to the end of a light elastic string, of unstretched length \(l\) and modulus of elasticity \(mgl/a\). The particle of mass \(2m\) is suddenly removed. Show that the other particle will come to rest again after a time \[ \left(\frac{2\pi}{3}+\sqrt{3}\right)\sqrt{\frac{a}{g}}. \]

\(A, B, C, P\) are four points in a plane. The line through \(A\) harmonically conjugate to \(AP\) with respect to the line pair \(AB, AC\) meets \(BC\) in \(L\); \(M\) and \(N\) are similarly defined on \(CA\) and \(AB\). Shew that \(L, M, N\) lie on a line \(p\) (the harmonic polar of \(P\) with respect to the triangle \(ABC\)). \par \(P\) moves on a conic \(S\) through \(A, B, C\). Prove that its harmonic polar passes through a fixed point \(O\), whose harmonic polar is its polar with respect to \(S\). \par \(A', B', C'\) are the points in which \(S\) is met again by the lines joining the vertices of the triangle \(ABC\) to the poles (with respect to \(S\)) of the opposite sides. Shew that the harmonic polar of any point of \(S\) with respect to the triangle \(A'B'C'\) also passes through \(O\).

Obtain a formula of reduction for \(\int \frac{\sin^m\theta}{\cos^n\theta}d\theta\), where \(m\) and \(n\) are positive and greater than 2. \par Shew that \[ \int_0^{\pi/4} \frac{\sin^6\theta}{\cos^8\theta}d\theta = \frac{5\pi}{8} - \frac{23}{12}, \] and evaluate \[ \int_0^{\pi/4} \frac{\sin^8\theta}{\cos^5\theta}d\theta. \]

Two uniform rods \(AB\), \(BC\) are of equal length and the weight of \(AB\) is \(n\) times that of \(BC\). They are freely jointed together at \(B\) and hang in equilibrium in a vertical plane with the ends \(A\) and \(C\) freely jointed to two pegs at the same level. Prove that the line of action of the reaction at the joint \(B\) passes through the point dividing \(AC\) externally in the ratio \(n:1\).

A coplanar system of forces acts on a rigid body. Shew that the system is equivalent either to a single force, or to a single couple. Find the conditions that the system is equivalent (i) to a force, (ii) to a couple. \par Referred to rectangular axes in the plane, the components of a typical force of the system are \((X_r, Y_r)\), and it acts at the point \((x_r, y_r)\), where \(r\) takes the values \(1, 2, \dots, n\). Find (i) the equation of the line of action of the force, if the system is equivalent to a force, (ii) the moment of the couple, if the system is equivalent to a couple.

A weight \(3w\) is supported by a tripod standing on the ground. Each leg of the tripod is of length \(l\) and weight \(w\), and the feet form an equilateral triangle of side \(a\). Equilibrium is maintained by three inextensible strings joining the middle points of the legs. Prove that if the ground is smooth, the tension in each string is \[ \frac{wa\sqrt{3}}{\sqrt{(3l^2-a^2)}}, \] but that, if the ground is rough, this tension may be reduced by \(4\mu w/\sqrt{3}\), where \(\mu\) is the coefficient of friction.