The dimensional formula of coefficient of shear modules is

Options

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

Answer

To find the dimensional formula of the coefficient of shear modulus, we will follow these steps:

### Step 1: Understand the definition of shear modulus
Shear modulus (often denoted as \( G \)) is defined as the ratio of shear stress to shear strain.

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

### Step 3: Define shear stress and shear strain
- **Shear Stress** is defined as force per unit area:
\[ \text{Shear Stress} = \frac{F}{A} \]
- **Shear Strain** is a dimensionless quantity, defined as the change in angle (in radians), which has no dimensions:
\[ \text{Shear Strain} = \frac{\Delta x}{L} \]
where \( \Delta x \) is the displacement and \( L \) is the original length.

### Step 4: Determine the dimensions of shear stress
- The dimensional formula for force \( F \) is:
\[ [F] = MLT^{-2} \]
- The dimensional formula for area \( A \) is:
\[ [A] = L^2 \]
Thus, the dimensional formula for shear stress becomes:
\[ [\text{Shear Stress}] = \frac{[F]}{[A]} = \frac{MLT^{-2}}{L^2} = ML^{-1}T^{-2} \]

### Step 5: Determine the dimensions of shear strain
Since shear strain is dimensionless:
\[ [\text{Shear Strain}] = 1 \]

### Step 6: Combine the dimensions to find the shear modulus
Now substituting the dimensions of shear stress and shear strain into the formula for shear modulus:
\[ [G] = \frac{[\text{Shear Stress}]}{[\text{Shear Strain}]} = \frac{ML^{-1}T^{-2}}{1} = ML^{-1}T^{-2} \]

### Conclusion
The dimensional formula of the coefficient of shear modulus is:
\[ [G] = ML^{-1}T^{-2} \]