Question

The dimensional formula of modulus of rigidity is
(a). $\left[ M{{L}^{2}}{{T}^{-2}} \right]$
(b). $\left[ M{{L}^{-1}}{{T}^{-3}} \right]$
(c).$\left[ M{{L}^{-2}}{{T}^{-2}} \right]$
(d). $\left[ M{{L}^{-1}}{{T}^{-2}} \right]$

Answer 1

- Hint: Modulus of rigidity is the ratio of the shear stress applied on the body to the shear strain caused on the body. First find the dimensional formula of shear stress and shear strain then you will be able to find the dimensional formula of modulus of rigidity.

Complete step-by-step solution -
When a force is applied on a body, there is deformation caused on the body and the body tends to resist this process by applying a force from within. The product of the force applied and the surface area on which the force is applied is called stress. Strain is the ratio of the change in the dimensions (size) of the body caused due to the applied force to the original value of that particular component (for example. length of a rubber band).
When the force applied on the surface of the body is parallel to the surface, the measured stress is called shear stress. The strain caused due to this stress is called shear strain. Here, the shear stress will be the product of the force applied and the parallel surface on which it is acting. When shear stress takes place, the opposite surfaces slide on one another and it causes a deformation in the angles of the corners as shown below.
If we talk about the shear strain, it is defined as the ratio of the displacement of the surface on which the force is applied to the length of the adjacent side.
To understand the relationship between shear stress and shear strain we have something known as the modulus of rigidity (G). Modulus of rigidity is the ratio of the shear stress to the shear strain.
Therefore, $G=\dfrac{\text{shear stress}}{\text{shear strain}}=\dfrac{F}{{A}}{\dfrac{\Delta l}{l}}=\dfrac{Fl}{\Delta lA}$
Where F is the force applied, A is the area tangential to the force, l is the length of the adjacent side and $\Delta l$ is the displacement of the surface.
Now calculate the dimensional formula of G.
$\left[ G \right]=\left[ \dfrac{Fl}{\Delta l A} \right]=\dfrac{\left[ ML{{T}^{-2}} \right]\left[ L \right]}{\left[ L \right]\left[ {{L}^{2}} \right]}=\left[ M{{L}^{-1}}{{T}^{-2}} \right]$
Therefore, the dimensional formula of modulus of rigidity is $\left[ M{{L}^{-1}}{{T}^{-2}} \right]$.
Hence, the correct option is (d).

Note: The value of G is always constant for a given material. It is necessary to note that the above statement is true within the elastic limit only. Elastic limit is the maximum point until where the stress is directly proportional to the strain. After the elastic limit, the stress will not be directly proportional to the strain.