Courses
Courses for Kids
Free study material
Offline Centres
More
Store

# The Young’s modulus of the material of a wire is $6 \times {10^{12}}N{m^{ - 2}}$ and there is no transverse in it, then its modulus of rigidity will be:A) $3 \times {10^{12}}N{m^{ - 2}}$B) $2 \times {10^{12}}N{m^{ - 2}}$C) ${10^{12}}N{m^{ - 2}}$D) $\text{None of the above}$

Last updated date: 14th Aug 2024
Total views: 70.5k
Views today: 0.70k
Verified
70.5k+ views
Hint: First, consider the relation between young’s modulus and modulus of rigidity of the material of the wire, i.e., $Y = 2\eta (1 + \sigma )$ . As no transverse is there in the wire, we can take $\sigma = 0$ . Now, from the above relation, calculate the modulus of rigidity $\eta$ .

Complete step by step solution:
From the relation between young’s modulus and rigidity modulus, we know that, $Y = 2\eta (1 + \sigma )$ ;
Where
$Y =$ Young’s modulus of the material of the wire,
$\eta =$ modulus of rigidity of the material of the wire,
$\sigma =$ transverse strain of the wire
By the given problem, there is no transverse strain in the wire, so $\sigma = 0$ .
So, we have $Y = 2\eta$ from the above relation.
Here, young’s modulus of the material of the wire is $Y = 6 \times {10^{12}}N{m^{ - 2}}$
Therefore, the value of modulus of rigidity of the material of the wire will be,
$\Rightarrow \eta = \dfrac{Y}{2}$
$\Rightarrow \dfrac{{6 \times {{10}^{12}}}}{2}N{m^{ - 2}}$
$\Rightarrow 3 \times {10^{12}}N{m^{ - 2}}$

The correct solution is (A), $3 \times {10^{12}}N{m^{ - 2}}.$

$Y = \dfrac{{F/A}}{{\Delta l/l}}$
$\eta = \dfrac{{F/A}}{{\tan \theta }}$
When $\theta$ is very small we have $\tan \theta \approx \theta$-
$\eta = \dfrac{{F/A}}{\theta }$
The relation between the two elastic constants is $Y = 2\eta (1 + \sigma )$ .
Note: Here, transverse strain $(\sigma )$ is the ratio of the change in diameter of a circular bar of a material to its diameter because of deformation in the longitudinal direction. It is also known as lateral strain. This quantity is dimensionless because of being a ratio between two quantities of the same dimension.