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Previous Years AIEEE/JEE Mains Questions
Q. A man grows into a giant such that his linear dimensions increase by a factor of 9. Assuming that his density remains same, the stress in the leg will change by a factor of :
(1) 81
( 2)$\frac{1}{81}$
(3) 9
(4) $\frac{1}{9}$
[JEE-Main-2017]
Ans.
Stress $=\frac{\text { Force }}{\text { area }}=\frac{\mathrm{mg}}{\mathrm{A}}=\frac{\text { volume } \times \text { density } \times \mathrm{g}}{\text { Area }}$
Stress $=\frac{\mathrm{L}^{3} \rho \mathrm{g}}{\mathrm{L}^{2}}$
Stress $\propto \mathrm{L}$
Q. A solid sphere of radius r made of a soft material of bulk modulus K is surrounded by a liquid in a cylindrical container. A massless piston of area a floats on the surface of the liquid, covering entire cross section of cylindrical container. When a mass m is placed on the surface of the piston to compress the liquid, the fractional decrement in the radius of the sphere, $\left(\frac{\mathrm{dr}}{\mathrm{r}}\right)$, is :
(1) $\frac{\mathrm{Ka}}{3 \mathrm{mg}}$
(2) $\frac{\mathrm{mg}}{3 \mathrm{Ka}}$
(3) $\frac{\mathrm{mg}}{\mathrm{Ka}}$
(4) $\frac{\mathrm{Ka}}{\mathrm{mg}}$
[JEE-Main-2018]
Ans.
$\left[\text { Bulk Modulus }=\frac{\text { volumetric stress }}{\text { volumetric strain }}\right]$
$\mathrm{K}=\frac{\mathrm{mg}}{\mathrm{a}\left(\frac{\mathrm{dV}}{\mathrm{V}}\right)}$
$\frac{\mathrm{d} \mathrm{V}}{\mathrm{V}}=\frac{\mathrm{mg}}{\mathrm{Ka}} \quad \ldots(\mathrm{i})$
volume of sphere $\rightarrow \mathrm{V}=\frac{4}{3} \pi \mathrm{R}^{3}$
Fractional change in volume $\frac{\mathrm{d} \mathrm{V}}{\mathrm{V}}=\frac{3 \mathrm{dr}}{\mathrm{r}} \ldots .$ (ii)
$\mathrm{U}$ sing eq. (i) $\&(2) \frac{3 \mathrm{dr}}{\mathrm{r}}=\frac{\mathrm{mg}}{\mathrm{Ka}}$
$\frac{\mathrm{dr}}{\mathrm{r}}=\frac{\mathrm{mg}}{3 \mathrm{Ka}}$