Two wires $A$ and $B$ of same material have radii in the ratio $2: 1$ and lengths in the ratio $4: 1$. The ratio of the normal forces required to produce the same change in the lengths of these two wires is .......

A compressive force, $F$ is applied at the two ends of a long thin steel rod. It is heated, simultaneously, such that its temperature increases by $\Delta T$. The net change in its length is zero. Let $l$ be the length of the rod, $A$ its area of cross- section, $Y$ its Young's modulus, and $\alpha $ its coefficient of linear expansion. Then, $F$ is equal to

There are two wire of same material and same length while the diameter of second wire is two times the diameter of first wire, then the ratio of extension produced in the wires by applying same load will be

Two similar wires under the same load yield elongation of $0.1$ $mm$ and $0.05$ $mm$ respectively. If the area of cross- section of the first wire is $4m{m^2},$ then the area of cross section of the second wire is..... $mm^2$

An equilateral triangle $ABC$ is formed by two copper rods $AB$ and $BC$ and one is aluminium rod which heated in such a way that temperature of each rod increases by $\Delta T$. Find change in the angle $\angle {ABC}$. (Coefficient of linear expansion for copper is $\alpha _1$ and for aluminium is $\alpha _2$).

Four uniform wires of the same material are stretched by the same force. The dimensions of wire are as given below. The one which has the minimum elongation has