Problems

(i) Determine \[ \lim_{x \to 0} \frac{\log(e^x+e^{-x}-1)}{\log \cos x}. \] (ii) Determine \[ \lim_{x \to \infty} \frac{e^{ax}-x}{e^{ax}+x} \] for all real values of \(a\).

Determine constants \(A, B, C, D\) such that \[ \frac{x^4+1}{(x^2+1)^4} = \frac{d}{dx} \frac{Ax^5+Bx^3+Cx}{(x^2+1)^3} + \frac{D}{x^2+1}. \] Hence or otherwise, prove that \[ 0.502 < \int_0^1 \frac{x^4+1}{(x^2+1)^4} dx < 0.503. \]

Find the volume of the body \[ (\sqrt{x^2+y^2}-a)^2 < b^2 - z^2 \] for \(0

If \(0

The equation \(z = F(x,y)\) is obtained by eliminating \(u\) between the equations \(y=f(u,x)\) and \(z=g(u,x)\). Prove that \[ \frac{\partial f}{\partial u} \frac{\partial F}{\partial x} = \frac{\partial f}{\partial u} \frac{\partial g}{\partial x} - \frac{\partial f}{\partial x} \frac{\partial g}{\partial u}, \] \[ \frac{\partial f}{\partial u} \frac{\partial F}{\partial y} = \frac{\partial g}{\partial u}. \]

Explain how necessary and sufficient conditions for the equilibrium of a coplanar system of forces can be expressed in terms of (i) Force Components, (ii) Moments. \par A smooth sphere of radius \(a\) and weight \(W\) is suspended from a fixed point \(O\) by a light inextensible string of length \(l\) attached to a point on its surface. Weights \(W_1\) and \(W_2\) are attached to smooth inextensible strings suspended from \(O\) and hanging over the surface of the sphere on opposite sides in the same vertical plane containing the first string. \par Determine the inclination of the first string to the vertical.

Explain what is meant by a couple acting on a body and define the moment of a couple. From your definitions establish the equivalence of coplanar couples of equal moment. \par A rough uniform rod of weight \(W\) lies on a horizontal table. The coefficient of friction is \(\mu\). \par Find the least value of a horizontal force applied at one end of the rod in a direction perpendicular to it which will disturb equilibrium.

A uniform heavy rod rests in equilibrium with its ends supported by rings which can slide on a rough circular wire held fixed in a vertical plane. The rod subtends an angle \(2\alpha\) at the centre. The angle of friction is \(\lambda\). Show that the greatest possible inclination of the rod to the horizontal is given by \(\cot\theta = \cot 2\alpha + \cos 2\alpha \csc 2\lambda\), provided the right-hand side is positive. Explain what conclusion is to be drawn if the right-hand side is either zero or negative.

Three rigid uniform rods \(AB, BC, CD\) are of unequal length and their weights are \(W, W'\) and \(W\) respectively. They are smoothly jointed at \(B\) and \(C\) and hang from smooth pivots \(A\) and \(D\) not necessarily on the same horizontal level. Shew that in the position of equilibrium the mid-point of \(BC\) is in its lowest possible position.

Two particles each of mass \(m\) are connected by a light inextensible string passing through a hole in a smooth horizontal table. One particle moves on the table; the other hangs below from the hole. When the lower particle is instantaneously at rest the other particle has speed \(v\) perpendicular to the string. Assuming that the string is of sufficient length, shew that in the subsequent motion the lower particle moves between two levels, one of which is the initial one.