The dimensions of shear modulus are

Options

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

Explanation

To find the dimensions of shear modulus (G), we start with the definition of shear modulus, which is the ratio of shear stress to shear strain.

### Step 1: Understand the definitions
- **Shear Stress (τ)** is defined as the force (F) applied per unit area (A):
\[ \tau = \frac{F}{A} \]

- **Shear Strain (γ)** is defined as the change in length (Δx) divided by the original length (L):
\[ \gamma = \frac{\Delta x}{L} \]

### Step 2: Write the formula for shear modulus
The shear modulus (G) can be expressed as:
\[ G = \frac{\text{Shear Stress}}{\text{Shear Strain}} = \frac{\tau}{\gamma} \]

### Step 3: Substitute the definitions into the formula
Substituting the definitions of shear stress and shear strain into the formula for shear modulus gives:
\[ G = \frac{\frac{F}{A}}{\frac{\Delta x}{L}} = \frac{F}{A} \cdot \frac{L}{\Delta x} \]

### Step 4: Rearrange the formula
This can be rearranged to:
\[ G = \frac{F \cdot L}{A \cdot \Delta x} \]

### Step 5: Determine the dimensions of each term
Now we need to find the dimensions of each term:
- **Force (F)** has dimensions:
\[ [F] = [M^1 L^1 T^{-2}] \]
- **Area (A)** has dimensions:
\[ [A] = [L^2] \]
- **Change in length (Δx)** has dimensions:
\[ [\Delta x] = [L^1] \]
- **Original length (L)** has dimensions:
\[ [L] = [L^1] \]

### Step 6: Substitute the dimensions into the formula
Now substituting these dimensions into the equation for G:
\[ [G] = \frac{[M^1 L^1 T^{-2}] \cdot [L^1]}{[L^2] \cdot [L^1]} = \frac{[M^1 L^2 T^{-2}]}{[L^3]} = [M^1 L^{-1} T^{-2}] \]

### Final Answer
Thus, the dimensions of shear modulus (G) are:
\[ [G] = [M^1 L^{-1} T^{-2}] \]