If the depression d at the end of a loaded bar is given by `d = (Mg l^3)/(3 yi)` where M is the mass , `l ` is the length and y is the young's modulus, then i has the dimensional formula

Options

A. `L^2`
B. `L^3 T`
C. `ML^3`
D. `L^4`

Answer

To find the dimensional formula for \( i \) from the equation given for the depression \( d \) at the end of a loaded bar:

\[ d = \frac{Mg l^3}{3y i} \]

where:
- \( M \) is the mass,
- \( g \) is the acceleration due to gravity,
- \( l \) is the length,
- \( y \) is the Young's modulus.

### Step 1: Identify the dimensions of each variable

1. **Depression \( d \)**: Since depression is a length, its dimensional formula is:
\[ [d] = L \]

2. **Mass \( M \)**: The dimensional formula for mass is:
\[ [M] = M \]

3. **Acceleration due to gravity \( g \)**: The dimensional formula for acceleration is:
\[ [g] = \frac{L}{T^2} \]

4. **Length \( l \)**: The dimensional formula for length is:
\[ [l] = L \]

5. **Young's modulus \( y \)**: Young's modulus has the same dimensions as pressure, which is force per unit area. The dimensional formula for pressure is:
\[ [y] = \frac{F}{A} = \frac{M L T^{-2}}{L^2} = \frac{M}{L T^2} \]

### Step 2: Substitute the dimensions into the equation

From the equation \( d = \frac{Mg l^3}{3y i} \), we can rearrange it to find \( i \):

\[ i = \frac{Mg l^3}{3d y} \]

Since the constant \( 3 \) does not affect the dimensional formula, we can ignore it.

### Step 3: Write the dimensions for \( i \)

Now substituting the dimensions we found:

\[ [i] = \frac{[M][g][l^3]}{[d][y]} \]

Substituting the dimensional formulas:

\[ [i] = \frac{M \cdot \frac{L}{T^2} \cdot L^3}{L \cdot \frac{M}{L T^2}} \]

### Step 4: Simplify the expression

Now let's simplify the expression step by step:

1. **Numerator**:
\[ M \cdot \frac{L}{T^2} \cdot L^3 = M \cdot L^4 \cdot T^{-2} \]

2. **Denominator**:
\[ L \cdot \frac{M}{L T^2} = \frac{M L}{L T^2} = \frac{M}{T^2} \]

Now substituting these back into the equation for \( i \):

\[ [i] = \frac{M L^4 T^{-2}}{\frac{M}{T^2}} = \frac{M L^4 T^{-2} \cdot T^2}{M} = L^4 \]

### Final Result

Thus, the dimensional formula for \( i \) is:

\[ [i] = L^4 \]