The mass per unit length of a non - uniform rod of length `L` is given `mu = lambda x^(2)` , where `lambda` is a constant and `x` is distance from one end of the rod. The distance of the center of mas of rod from this end is

Options

A. `(L)/(2)`
B. `(L)/(4)`
C. `(3L)/(4)`
D. `(L)/(3)`

Explanation

To find the center of mass of a non-uniform rod with a mass per unit length given by \( \mu = \lambda x^2 \), where \( \lambda \) is a constant and \( x \) is the distance from one end of the rod, we can follow these steps:

### Step 1: Define the mass element
The mass per unit length is given as:
\[ \mu = \lambda x^2 \]
To find the mass of a small segment \( dx \) of the rod at a distance \( x \) from one end, we can express the mass \( dm \) of this segment as:
\[ dm = \mu \cdot dx = \lambda x^2 \cdot dx \]

### Step 2: Set up the center of mass formula
The center of mass \( x_{cm} \) of the rod can be calculated using the formula:
\[ x_{cm} = \frac{\int x \, dm}{\int dm} \]
Substituting the expression for \( dm \):
\[ x_{cm} = \frac{\int x \cdot (\lambda x^2 \, dx)}{\int (\lambda x^2 \, dx)} \]

### Step 3: Calculate the numerator and denominator
**Numerator:**
\[ \int x \cdot (\lambda x^2 \, dx) = \lambda \int x^3 \, dx \]
Calculating this integral from \( 0 \) to \( L \):
\[ \lambda \int_0^L x^3 \, dx = \lambda \left[ \frac{x^4}{4} \right]_0^L = \lambda \frac{L^4}{4} \]

**Denominator:**
\[ \int (\lambda x^2 \, dx) = \lambda \int_0^L x^2 \, dx \]
Calculating this integral from \( 0 \) to \( L \):
\[ \lambda \int_0^L x^2 \, dx = \lambda \left[ \frac{x^3}{3} \right]_0^L = \lambda \frac{L^3}{3} \]

### Step 4: Substitute back into the center of mass formula
Now substituting the results from the numerator and denominator back into the center of mass formula:
\[ x_{cm} = \frac{\lambda \frac{L^4}{4}}{\lambda \frac{L^3}{3}} = \frac{L^4}{4} \cdot \frac{3}{L^3} = \frac{3L}{4} \]

### Final Result
Thus, the distance of the center of mass of the rod from one end is:
\[ \boxed{\frac{3L}{4}} \]