A rod of length 3 m has its mass per unit length directly proportional to distance x from one of its ends then its centre of gravity from that end will be at

Options

A. 1.5 m
B. 2 m
C. 2.5 m
D. 3.0 m

Answer

To find the center of gravity of a rod of length 3 m, where the mass per unit length is directly proportional to the distance from one end, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Problem**:
We have a rod of length \( L = 3 \) m. The mass per unit length \( \lambda \) is given to be directly proportional to the distance \( x \) from one end of the rod. Thus, we can express this as:
\[ \lambda = kx \]
where \( k \) is a constant of proportionality.

2. **Consider a Small Element**:
Take a small element of the rod of length \( dx \) at a distance \( x \) from the chosen end. The mass \( dm \) of this small element can be expressed as:
\[ dm = \lambda \cdot dx = kx \cdot dx \]

3. **Calculate Total Mass**:
The total mass \( M \) of the rod can be calculated by integrating \( dm \) from \( 0 \) to \( 3 \) m:
\[ M = \int_0^3 dm = \int_0^3 kx \, dx \]
Evaluating this integral:
\[ M = k \int_0^3 x \, dx = k \left[ \frac{x^2}{2} \right]_0^3 = k \cdot \frac{3^2}{2} = \frac{9k}{2} \]

4. **Calculate the Center of Gravity**:
The center of gravity \( x_{cm} \) is given by the formula:
\[ x_{cm} = \frac{1}{M} \int_0^3 x \, dm \]
Substituting \( dm = kx \, dx \):
\[ x_{cm} = \frac{1}{M} \int_0^3 x \cdot (kx) \, dx = \frac{k}{M} \int_0^3 x^2 \, dx \]
Evaluating the integral:
\[ \int_0^3 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^3 = \frac{3^3}{3} = 9 \]
Thus,
\[ x_{cm} = \frac{k \cdot 9}{M} \]

5. **Substituting for M**:
Now substituting \( M = \frac{9k}{2} \) into the expression for \( x_{cm} \):
\[ x_{cm} = \frac{k \cdot 9}{\frac{9k}{2}} = \frac{9 \cdot 2}{9} = 2 \]

6. **Final Result**:
Therefore, the center of gravity of the rod from the chosen end is:
\[ x_{cm} = 2 \text{ m} \]

### Conclusion:
The center of gravity of the rod is located at a distance of **2 meters** from the end where the mass per unit length is measured.