Problems

A particle of mass \(m\) is suspended from a fixed point by a light string which is blown from the vertical by a steady horizontal wind of uniform velocity \(V\). Assuming that the force exerted by the wind on each element of length \(\delta s\) of the string is normal to the element and of magnitude \(\kappa \delta s \, v^2\), \(v\) being the component of \(V\) normal to the element, shew that the string hangs in a catenary. If the wind pressure on the particle is negligible, prove that the depth of the particle below the point of suspension is \[ c \log\left[\frac{l+\sqrt{l^2+c^2}}{c}\right], \] where \(c=mg/KV^2\), and \(l\) is the length of the string.

Shew that the reciprocal of a conic, with respect to a focus \(S\), is a circle, and that, if the conic is a parabola, the circle passes through \(S\). A variable parabola touches a fixed conic and has its focus at one of the foci of the conic. Prove that its directrix touches a fixed circle.

Prove that if \(\tan \alpha, \tan \beta, \tan \gamma\) are in arithmetic progression, then so are \(\cot (\alpha - \beta)\), \(\tan \beta\), \(\cot (\gamma - \beta)\).

Prove that, if \(u_n = \int_0^\pi \frac{dx}{(a+b\cos x+c\sin x)^n}\), then for integral values of \(n\) \[ (n-1)(a^2-b^2-c^2)u_n - (2n-3)au_{n-1} + (n-2)u_{n-2} + c\left\{\frac{1}{(a-b)^{n-1}} + \frac{1}{(a+b)^{n-1}}\right\} = 0. \] Obtain the values of \(u_1\) and \(u_2\) when \(a^2>b^2+c^2\), \(a\) being supposed positive.

A uniform beam of length \(2l\) rests symmetrically on two supports which are a distance \(2a\) apart in a horizontal line; prove that the beam is least liable to break if \(a=l(2-\sqrt{2})\), it being assumed that the beam is liable to break if a definite bending moment is exceeded at any point.

Find the harmonic conjugate of the line \(y=mx\) with respect to the pair of lines \[ ax^2 + 2hxy + by^2 = 0. \] A variable line through the fixed point \((\alpha, \beta)\) cuts the lines \[ ax^2 + 2hxy + by^2 = 0 \] in \(P\) and \(Q\). Prove that the locus of the mid-point of \(PQ\) is the conic \[ (ax+hy)(x-\alpha) + (hx+by)(y-\beta) = 0. \]

Prove that the line joining the circumcentre and the orthocentre of the triangle \(ABC\) makes with \(BC\) the angle \[ \tan^{-1}\left(\frac{\tan B \tan C - 3}{\tan B - \tan C}\right). \]

A uniform sphere of weight \(W\) rests on a horizontal plane touching it at \(C\). A uniform beam \(AB\) of weight \(X\) has its end \(A\) on the plane and is a tangent to the sphere at \(B\), \(ABC\) being a vertical plane. The ratios of the tangential to the normal reactions at \(A,B,C\) in equilibrium are \(\alpha, \beta, \gamma\). Shew that \(\beta\) is greater than \(\alpha\) and than \(\gamma\), and that \(\alpha > \gamma\) if \(W > X(1-\cos BAC)\).

A horse pulls a cart starting from rest at \(A\); the pull exerted gradually decreases until on reaching \(B\) it is equal to the constant resistance to motion due to friction, etc. Shew that, if the decrease in pull is proportional to the time from leaving \(A\), the velocity at \(B\) is \(\frac{\sqrt{3}}{2}\) of the velocity at \(B\) if the decrease is proportional to the distance from \(A\), the initial pull being the same in each case.

Shew that the equation \[ x^2+y^2+2gx+c=0 \] represents, for a given \(c\) and different \(g\)'s, a system of coaxal circles. Prove that the poles of a fixed line \(l\) with respect to the circles of a coaxal system lie on a fixed conic \(S\). Shew that, if \(l\) is not parallel to the radical axis or to the line of centres and does not pass through a limiting point of the system, then \(S\) is a (non-degenerate) hyperbola whose asymptotes are given by the following construction. Let \(l\) meet the line of centres and the radical axis in \(P\) and \(Q\) respectively; take \(R\) on \(l\) so that \(QR=PQ\); then the asymptotes are the lines through \(P\) and \(R\) perpendicular to \(l\) and parallel to the radical axis respectively.