Problems

Find the rhombus of maximum area and the rhombus of minimum area inscribed in the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

Shew that a system of coplanar forces may be reduced (i) to a force acting at an assigned point \(P\) in the plane of the forces together with a couple \(G\), or (ii) to three forces acting along the sides of a given triangle \(ABC\) in the plane of the forces. What is the locus of \(P\) for a given value of \(G\)? If this locus passes through \(A\) and through the mid-point of \(BC\), and if the forces in (ii) are \(X\) in direction \(\vec{BC}\), \(5Z\) in direction \(\vec{CA}\), \(Z\) in direction \(\vec{AB}\), determine \(X\) in terms of \(Z\), it being given that the lengths of \(BC, CA, AB\) are in the ratio \(9:5:8\).

A wedge of mass \(M\) has two smooth plane faces inclined at an angle \(\alpha\), and is placed with one face in contact with a smooth horizontal table. A particle of mass \(m\) is placed centrally on the other face and allowed to slide down under gravity. Find the acceleration of the wedge and the force it exerts on the table.

Prove that an ellipse has two equal conjugate diameters. Shew further that the locus of the point of intersection of normals at the extremities of chords parallel to one of these is a straight line perpendicular to the other.

Prove that, if \(c_n\) is the coefficient of \((x+1)^n\) in the expansion of \[ \frac{e^{x^2+2x}}{(x^2+2x+2)^2} \] in a series of positive powers of \((x+1)\), then \(c_n=0\) if \(n\) is odd, while \[ c_{2k} = \frac{1}{2e} \sum_{m=0}^k \frac{(-1)^m(m+1)(m+2)}{(k-m)!} \quad (k=0, 1, 2, \dots). \]

\(a\) is the unstretched length and \(kmg\) the modulus of elasticity of a light extensible string, to one end of which is attached a particle of mass \(m\). The other end \(O\) of the string is fixed to a rough plane inclined to the horizontal at an angle \(\alpha\) which exceeds the angle of friction, \(\lambda\), between the particle and the plane. If the particle is also at \(O\) and is released from rest, find an expression for the distance through which it will slide before its velocity is again zero. Shew further that the particle will remain at rest in this new position unless \(3\tan\lambda\) is less than \[ (k^2\sec^2\alpha+8k\sec\alpha\tan\alpha+4\tan^2\alpha)^{\frac{1}{2}} - k\sec\alpha - \tan\alpha. \]

A smooth ball of mass \(m\) hangs at rest on a light inextensible string from a fixed point. A second smooth ball of mass \(m'\) impinges directly on the first so that its velocity \(V\) makes an acute angle \(\alpha\) with the string. The coefficient of restitution between the two balls is \(e\). Shew that the initial velocity of the first ball is \[ \frac{m'(1+e)V\sin\alpha}{m+m'\sin^2\alpha}, \] and that the impulse in the string is \[ \frac{mm'(1+e)V\cos\alpha}{m+m'\sin^2\alpha}. \]

Shew that a circle drawn through the centre of a rectangular hyperbola and any two points will also pass through the intersection of lines drawn through each of these points parallel to the polar of the other.

Prove that the polynomial \(X_n = \frac{d^n}{dx^n}(x^2-1)^n\) satisfies the equation \[ (1-x^2)\frac{d^2X_n}{dx^2} - 2x \frac{dX_n}{dx} + n(n+1)X_n = 0. \]

Obtain expressions for the tangential and normal components of the acceleration of a particle moving in a plane curve. A bead is free to move on a smooth wire in the form of a catenary fixed with its axis vertically upwards. The bead is projected from the lowest point of the catenary and, in the subsequent motion, the ratio of the greatest and least values of the reaction between the bead and the wire is 15. Determine the least value of the reaction in terms of the weight of the bead.