Two particles of masses \(m\) and \(M\) are connected by a light elastic string of natural length \(l\) and modulus of elasticity \(\lambda\). Shew that it is possible for the particles to describe concentric circles with uniform angular velocities each equal to \(\omega\), the distance between them being \[ \frac{(m+M)l\lambda}{(m+M)\lambda - mMl\omega^2}; \] and that the period of small oscillations about the state of steady motion is \(\pi/\omega\).
Calculate the principal moments of inertia at the vertex of a uniform right circular cone of semivertical angle \(\alpha\) and of mass \(M\). Such a cone has its vertex \(O\) and a point \(P\) on the rim of its base fixed, and rotates under the action of no forces with angular velocity \(\omega\) about \(OP\). Shew that the stress at \(O\) is \(\frac{1}{4}Mr\omega^2(5\sin^2\alpha+1)\), and at \(P\) is \(\frac{1}{4}Mr\omega^2(5\cos^2\alpha-1)\); where \(r\) is the radius of the circle described by the centroid of the cone.
A light string of length \(6l\) is stretched between two fixed points with tension \(T\); two particles, each of mass \(m\), are attached at the points of trisection, and a particle of mass \(M\) at the middle point. Shew that in small transverse oscillations one period is \(2\pi\sqrt{\left(\frac{2}{3}\frac{ml}{T}\right)}\); and that the other two periods cannot lie between this value and \(2\pi\sqrt{\left(\frac{Ml}{2T}\right)}\).
A sphere of S.I.C. \(K\) is introduced into a uniform field of electric force. Obtain expressions for the electric potential at points inside and outside the sphere; and shew that the greatest discontinuity in the direction of a line of force at the surface of the sphere is \[ \frac{\pi}{2} - 2(\text{arc cot}\sqrt{K}). \]
A magnetic molecule is placed along the axis of a circular conductor of radius \(a\) at a point where any radius of the circle subtends an angle \(\alpha\). The magnetic moment of the molecule is periodic and equal to \(\mu\cos pt\). Shew that the periodic current in the conductor is \[ \frac{2\pi\mu p\sin^3\alpha}{a} \frac{pL\cos pt - R\sin pt}{p^2L^2+R^2}. \] Find the mechanical force on the conductor at any time, and prove that its mean value is \[ \frac{6\pi^2\cos\alpha\sin^4\alpha}{a^3} \frac{\mu^2 p^2 L}{p^2L^2+R^2}; \] where \(L,R\) are the coefficient of self-induction and resistance of the conductor.
The feet of the perpendiculars from a point \(P_1\) to the sides of a triangle \(ABC\) lie on a straight line \(\lambda_1\); prove that \(P_1\) lies on the circle circumscribing the triangle \(ABC\). Prove that, if \(P_2, \lambda_2\) and \(P_3, \lambda_3\) are related in the same way, the triangle \(P_1P_2P_3\) is similar to the triangle formed by \(\lambda_1, \lambda_2, \lambda_3\).
Prove that the operations of inversion with respect to two coplanar circles in succession are commutative if the circles cut one another orthogonally. \(A', B', C'\) are the inverses of three fixed points \(A, B, C\) with regard to a variable circle whose centre is \(P\). Find the locus of \(P\) when the triangle \(A'B'C'\) is right angled.
Given a focus and the corresponding directrix of a conic and also the eccentricity, obtain a geometrical construction for the two points in which the conic is cut by any straight line.
A variable line moves in a plane so that the intercepts made on it by the sides of a fixed coplanar triangle bear constant ratios to one another. Show that the line envelopes a parabola inscribed in the fixed triangle.
A point moves on a given plane so that the line joining it to a fixed point not in the plane makes a given angle with a fixed line in the plane. Show that the locus of the point is a hyperbola and that as the given angle varies the corresponding hyperbolas are coaxal.