The dimensional formula for modulus of rigidity is

Options

A. M L T
B. `ML^(-2)T^(3)`
C. `ML^(-1)T^(-1)`
D. `ML^(-1)T^(-2)`

Answer

To find the dimensional formula for the modulus of rigidity, we can follow these steps:

### Step 1: Understand the Definition
The modulus of rigidity (μ) is defined as the ratio of shear stress to shear strain.

### Step 2: Write the Formula
The formula for modulus of rigidity is given by:
\[ \mu = \frac{\text{Shear Stress}}{\text{Shear Strain}} \]

### Step 3: Determine the Dimensions of Shear Stress
Shear stress is defined as force per unit area. The dimensional formula for force (F) is:
\[ [F] = MLT^{-2} \]
The area (A) has the dimensional formula:
\[ [A] = L^2 \]
Thus, the dimensional formula for shear stress (τ) is:
\[ [\tau] = \frac{[F]}{[A]} = \frac{MLT^{-2}}{L^2} = ML^{-1}T^{-2} \]

### Step 4: Determine the Dimensions of Shear Strain
Shear strain is defined as the change in length divided by the original length (ΔL/L). Since both ΔL and L are lengths, shear strain is a dimensionless quantity:
\[ [\text{Shear Strain}] = 1 \quad (\text{dimensionless}) \]

### Step 5: Combine the Dimensions
Now we can substitute the dimensions of shear stress and shear strain into the formula for modulus of rigidity:
\[ [\mu] = \frac{[\text{Shear Stress}]}{[\text{Shear Strain}]} = \frac{ML^{-1}T^{-2}}{1} = ML^{-1}T^{-2} \]

### Conclusion
Thus, the dimensional formula for the modulus of rigidity is:
\[ \mu = ML^{-1}T^{-2} \]

### Final Answer
The dimensional formula for modulus of rigidity is \( ML^{-1}T^{-2} \).

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