Problems

\(P\) is a variable point of a conic \(S\), and \(Q\) is the centre of the rectangular hyperbola having four-point contact with \(S\) at \(P\). If \(S\) is a circle, show that as \(P\) varies \(Q\) describes a concentric circle of twice the radius. If \(S\) is a parabola, show that the locus of \(Q\) is an equal parabola with the same axis and directrix as \(S\).

Prove that the maxima of the curve \(y=e^{-kx}\sin px\) (\(k\) and \(p\) being positive constants) all lie on a curve whose equation is \(y=Ae^{-kx}\), and find \(A\) in terms of \(k\) and \(p\). Draw in the same diagram rough sketches of the curves \(y=e^{-kx}\), \(y=-e^{-kx}\) and \(y=e^{-kx}\sin px\) for positive values of \(x\).

A light inextensible string of length \(2l\) is fastened at one end to a fixed point; it carries a mass \(m\) at the mid-point and a mass \(2m\) at the lower end. The system is slightly disturbed from rest so that the masses move in the same vertical plane. If the horizontal displacements of the upper and lower masses are \(x_1\) and \(x_2\) respectively (both being measured in the same direction) show that the equations of motion of the masses are \begin{align*} \frac{d^2 x_1}{dt^2} + n^2 (5x_1 - 2x_2) &= 0, \\ \frac{d^2 x_2}{dt^2} + n^2 (x_2 - x_1) &= 0, \end{align*} where \(n^2=g/l\), and terms of higher order are neglected. Show that a state of oscillation is possible in which the masses execute simple harmonic motions of the same period, with \(x_1 = x_2(\sqrt{6}-2)\), and find the period of oscillation.

Explain fully what is meant by the dimensions of a physical quantity. The measure of a certain physical quantity is found to be 1 when pound-foot-second units are used, \(16\) when ounce-foot-second units are used, \(9\) when ounce-inch-second units are used, and \(h\) when ton-mile-hour units are used. Find its dimensions in mass, length and time, and compare the unit of this physical quantity in the ounce-inch-second system with that in the ton-mile-hour system.

The equations \(S=0\), \(u=0\) and \(v=0\) represent respectively a conic and two straight lines. Interpret the equations \(S+\lambda uv = 0\), \(S+\lambda u^2=0\), where \(\lambda\) is a parameter. Two conics \(S_1\) and \(S_2\) each have double contact with \(S\). Show that two of the common chords of \(S_1\) and \(S_2\) meet at the point of intersection of the chords of contact of \(S_1\) and \(S_2\) with \(S\), and form with them a harmonic pencil.

Evaluate the integrals \[\int_1^2 \{\sqrt{(2-x)(x-1)}\} \,dx, \quad \int_0^\infty (1+x^2)^2 e^{-x} \,dx.\]

Forces of magnitudes \(m\text{OA}\), \(n\text{OB}\) act in the lines OA, OB respectively. Prove that the resultant force is \((m+n)\text{OC}\), where C is the point in AB such that \(m\text{AC} = n\text{CB}\). Forces \(4P, -8P, 6P, -3P\) act in the sides \(AB, BC, CD, DE\) respectively of a regular hexagon \(ABCDEF\). Find the magnitude and line of action of the resultant force.

A uniform solid rectangular block, of edges \(2a, 2b, 2c\), rests on an inclined plane, the coefficient of friction between the block and the plane being \(a/b\). The edges of length \(2a\) are parallel to the lines of greatest slope, and the edges of length \(2c\) are horizontal. A string is fastened to the block at the middle point of the highest edge of length \(2c\), and is parallel to a line of greatest slope. If the tension in the string is gradually increased, tending to pull the block up the plane, determine whether equilibrium is broken by tilting or by slipping.

Find the centre of gravity of a thin uniform hemispherical bowl. A uniform hemispherical bowl is bounded by two concentric spheres of radii \(a, b\) (\(b>a\)) and a diametral plane. Shew that the distance of the centre of gravity of the bowl from the common centre of the spheres is \(3(b^4-a^4)/8(b^3-a^3)\). A particle of mass \(m\) is fixed to the bowl at a point of the boundary common to the diametral plane and the sphere of radius \(b\), and the bowl rests in equilibrium with its curved surface on a smooth horizontal table. Prove that the axis of symmetry of the bowl makes an angle \(\theta\) with the vertical, where \[ \tan\theta = 4mb/\pi\rho(b^4-a^4), \] \(\rho\) being the density of the bowl.

Five light rods \(AB, BC, CD, DE, EF\), each of length \(2a\), are freely hinged at \(B, C, D, E\) and a light string, also of length \(2a\), joins \(A\) and \(F\). The chain so formed is wrapped round a smooth circular cylinder of radius \(a\sqrt{3}\), so that the figure assumes the form of a regular hexagon. If the string is drawn to tension \(T\), find the reactions at the hinges.