A bob of a simple pendulum of mass 'm' is displaced through 90$$^\circ$$ from mean position and released. When the bob is at lowest position, the tension in the string is

Let '$$\mathrm{W}_1$$' be the work done in blowing a soap bubble of radius '$$r$$' from soap solution at room temperature. The soap solution is now heated and second soap bubble of radius '$$2 r$$' is blown from the heated soap solution. If '$$W_2$$' is the work done in forming this bubble then

A cylindrical rod is having temperatures $$\theta_1$$ and $$\theta_2$$ at its ends. The rate of heat flow is '$$Q$$' $$\mathrm{J}{\mathrm{s}}^{-1}$$. All the linear dimensions of the rod are doubled by keeping the temperatures constant. What is the new rate of flow of heat?

Two concentric coplanar circular loops of radii '$$r{ }_1$$' and '$$r_2$$' respectively carry currents '$$i_1$$' and '$$\mathrm{i}_2$$' in opposite directions (one clockwise and other anticlockwise). The magnetic induction at the centre of the loops is half that due to '$$i_1$$' alone at the centre. If $$r_2=2 r_1$$, the value of $$\frac{i_2}{i_1}$$