Question

The ratio of lengths of two rods A and B of same material is $1:2$ and the ratio of their radii is $2:1$ , then the ratio of rigidity of A and B will be
A. $4:1$
B. $16:1$
C. $8:1$
D. $1:1$

Answer 1

Hint:Apply the formula of the modulus of rigidity to the given problem in order to obtain an appropriate solution since we know that the modulus of rigidity plays a vital role in determining the relationship between shear stress and shear strain in problems based on the mechanical properties of solids.




Formula used:
$Modulus{\text{ }}of{\text{ }}Rigidity(\eta ) = \dfrac{{ShearStress({\sigma _s})}}{{ShearStrain(\theta )}}$




Complete answer:
We know that $Modulus{\text{ }}of{\text{ }}Rigidity(\eta ) = \dfrac{{ShearStress({\sigma _s})}}{{ShearStrain(\theta )}}$
Now, the ratio of lengths of two rods A and B is $\dfrac{{{l_A}}}{{{l_B}}} = \dfrac{1}{2}$ (given)
And the ratio of their radii is $\dfrac{{{r_A}}}{{{r_B}}} = \dfrac{2}{1}$ (given)
Also, we know that the modulus of rigidity $(\eta )$ is the property of the material and it changes when the nature of material changes. But, as the two rods are of the same material as given in the question, therefore, the modulus of rigidity will be the same for both the rods i.e., the ratio will be: -
$ \Rightarrow \dfrac{{{\eta _A}}}{{{\eta _B}}} = \dfrac{1}{1}$
Thus, the ratio of modulus of rigidity of A and B is $1:1$ .
Hence, the correct option is (D) $1:1$.


Therefore, the answer is option (D)



Note: The ratio of shear stress to the shear strain of a mechanical member is known as the modulus of rigidity or shear modulus. The member's material will determine this attribute; the more elastic the member, the greater the modulus of rigidity.