Problems

The portion of the curve \(y=f(x)\) included between the ordinates \(x=a\) and \(x=b\) (\(a < b\)) is rotated about the axis of \(x\). Prove that the volume of the surface of revolution so obtained is \[ V = \pi \int_a^b \{f(x)\}^2 dx, \] and that the area of the curved surface is \[ S = 2\pi \int_a^b f(x) \{1+(f'(x))^2\}^{\frac{1}{2}} dx, \] where \(f'(x) = \dfrac{d}{dx}f(x)\). Find the volume and area of the surface of the solid obtained by rotating the portion of the cycloid \[ x=a(\theta+\sin\theta), \quad y=a(1+\cos\theta) \] between two consecutive cusps about the axis of \(x\).

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, stating clearly the assumptions which are needed in the course of the proof. Shew further that the work done by the system in an arbitrary infinitesimal displacement of the body is equal to the work done by the equivalent force or couple.

A uniform lamina in the shape of an equilateral triangle \(ABC\) of side \(a\) is free to move in a vertical plane, the edges \(AB, BC\) resting on two smooth pegs \(P, Q\) at the same level and at distance \(b\) apart. If \(G\) is the centroid of the lamina, shew that, whatever the inclination of the lamina, \(BG\) passes through a fixed point. Shew also that the height of \(G\) above this fixed point when \(BG\) makes an angle \(\theta\) with the vertical is \[ (a\cos\theta-b-b\cos 2\theta)/\sqrt{3}. \] Hence shew that the position of equilibrium with \(BG\) vertical is stable if \(a < 4b\), and unstable if \(a > 4b\). Is the equilibrium stable or unstable if \(a=4b\)?

A particle moves in a straight line under the action of a given (variable) force. What physical quantity is represented by the area lying under the curve and bounded by two ordinates in the several cases when the abscissae and ordinates represent graphically (1) time and velocity, (2) time and force, (3) distance and force? A cable is used for raising loads, the greatest tension that the cable will bear being \(W\) tons weight. Shew, by consideration of the time-velocity graph, that the least time in which a load of \(W'\) tons can be raised through \(h\) feet from rest to rest by means of the cable is \(\sqrt{\left\{\dfrac{2h W}{g(W-W')}\right\}}\) seconds. The weight of the cable itself is negligible.

Establish the formulae \(dv/dt, v^2/\rho\), for the tangential and normal components of acceleration of a point moving on a given curve, where \(v\) denotes the velocity of the moving point, and \(\rho\) the radius of curvature of the curve. A light inextensible string \(AB\) of length \(l\) has the end \(A\) attached to a point of the surface of a fixed cylinder, whose cross-section is a simple closed oval curve whose intrinsic equation is \(s=f(\psi)\). Both \(\psi\) and \(s\) vanish at \(A\), and \(s\) increases always with \(\psi\). A particle of mass \(m\) is attached to \(B\), and is acted on by a constant force \(mc\) at right angles to the string, so that the string wraps itself round the cylinder, the whole motion being in a plane at right angles to the generators. Find the relation connecting the time with the inclination \(\psi\) of the straight part of the string to the tangent at \(A\), and shew that the tension of the string is \[ m \frac{u^2+2c\{l\psi - F(\psi)\}}{l-f(\psi)}, \] where \(u\) is the velocity of the particle when \(\psi=0\), and \(F(\psi)=\int_0^\psi f(x)dx\).

A particle of mass \(m\) moves in a plane, and is attracted towards a fixed origin \(O\) in the plane with a force \(mn^2r\), where \(r\) denotes distance from \(O\). It is projected from the point \((c,0)\), the axes being rectangular, with velocity \(nb\) and in a direction inclined at an angle \(\theta\) to the axis \(Ox\). Shew that the path of the particle is the ellipse \[ b^2(x\sin\theta-y\cos\theta)^2 + c^2y^2 = b^2c^2\sin^2\theta. \] Shew further that the points of the plane which are accessible by projection from the given point with the given velocity lie within the ellipse \[ \frac{x^2}{b^2+c^2} + \frac{y^2}{b^2} = 1. \]

Shew that four normals to an ellipse can be drawn through a general point of its plane. Shew that the normals at three points whose eccentric angles are \(\theta_1, \theta_2, \theta_3\) meet if \[ \sin(\theta_2+\theta_3) + \sin(\theta_3+\theta_1) + \sin(\theta_1+\theta_2) = 0, \] and that the fourth normal through their common point is the normal at the point whose eccentric angle is \(\theta_4\), where \(\theta_1+\theta_2+\theta_3+\theta_4\) is an odd multiple of \(\pi\).

If \(r\) denotes the distance of a point \(Q\) lying on a given curve from a fixed point \(S\) in the plane of the curve, and \(p\) is the perpendicular distance from \(S\) to the tangent at \(Q\) to the given curve, shew that the radius of curvature at \(Q\) is \(r\dfrac{dr}{dp}\). If the given curve is an ellipse of semi-axes \(a\) and \(b\) (\(a>b\)) and \(S\) is a focus, shew that \[ \frac{l}{p^2} = \frac{2}{r} - \frac{1}{a}, \quad \text{where } l=b^2/a, \] and hence determine its maximum and minimum radii of curvature.

Sketch the curve \[ x^3 = 3xy^2 + a^2x + y^2. \] Trace the inverse of the curve in the circle \[ x^2+y^2=1, \] and find the area of a loop of this inverse.

Explain what is meant by saying that a certain event has probability \(r\) (\(0 \le r \le 1\)). \(X\) and \(Y\) are partners at bridge against \(A\) and \(B\). \(X\) is dummy and when he puts his hand on the table \(Y\) sees that six trumps are held by the opponents. Shew that the probability that \(A\) and \(B\) each hold three trumps is \(\dfrac{286}{805}\).