Question

A solid sphere hung at the lower end of a wire is suspended from a fixed point so as to give an elongation of $0.4 \mathrm{~mm} .$ When the first solid sphere is replaced by another one made of same material but twice the radius, the new elongation is
$\text{1) 0}\text{.8}~\text{mm}$
$\text{2) }1.6~\text{mm}$
$\text{3) }3.2~\text{mm}$
$\text{4) }1.2~\text{mm}$

Answer 1

Hint: The elongation is measured as the duration of the relative increase. Break elongation is measured in percent (percent of elongation vs. original size when there is a break). The full elongation, i.e., emax, is also called "strain to failure" at split. Calculate Stress and strain of the sphere and use the relation Stress is directly proportional to strain.
Formula used: $Stress=\dfrac{Force}{4\pi {{r}^{2}}}$

Complete answer:
A sensation of emotional or physical tension is stress. Any event or thought that makes you feel upset, angry, or anxious may come from it. Stress is the response of the body to a threat or demand. Stress can be beneficial in brief bursts, such as when it helps you escape danger or reach a deadline.
By law,
Stress is directly proportional to strain
$Stress=\dfrac{Force}{4\pi {{r}^{2}}}$
$Strain=\dfrac{\vartriangle L}{l}$
 On Substituting the values
We get,
For first sphere before the change
$\dfrac{F}{4\pi {{r}^{2}}}\propto ~\dfrac{\Delta l}{l}$
\[\Rightarrow \dfrac{4\pi \rho {{\text{r}}^{3}}~\text{g}}{3\times 4\pi {{r}^{2}}}\propto ~\dfrac{0.4}{1}\]
$\dfrac{\rho rg}{3}\propto ~\dfrac{0.4}{l}$ - (I)
For second sphere after the change
${{r}^{\prime }}=2r$
$\Rightarrow \dfrac{2\rho rg}{3}\propto ~\dfrac{\Delta l}{I}$ -(ii)
Divide equation (II) By (II)
We get,
$\Rightarrow \dfrac{\dfrac{2\rho rg}{3}\propto ~\dfrac{\Delta l}{I}}{\dfrac{\rho rg}{3}\propto ~\dfrac{0.4}{l}}$
$\Rightarrow \dfrac{2}{1}=\dfrac{\Delta l}{0.4}$
$\Delta I=0.8 \mathrm{~mm}$
$\therefore $Therefore, the new elongation is $0.8 \mathrm{~mm}$.

The correct option is (1).

Note:
On a vertical diameter plane, the stresses in a spherical vessel cause one to cut through the sphere and separate half of the shell and its fluid contents as a single free body. The tensile tension σ in the vessel wall and the fluid pressure p. are working on this free body. The portion of the total stress corresponding to the isotropic hydrostatic pressure; the unit tensor is its stress tensor, multiplied by one-third the total stress tensor trace.