If linear density of a rod of length 3m varies as `lambda = 2 + x,` them the position of the centre of gravity of the rod is

Options

A. `(7)/(3)m`
B. `(12)/(7)`
C. `(10)/(7)m`
D. `(9)/(7)m`

Answer

To find the position of the center of gravity of a rod of length 3 m with a linear density that varies as \( \lambda = 2 + x \), we can follow these steps:

### Step 1: Define the Linear Density
The linear density of the rod is given by:
\[ \lambda(x) = 2 + x \]
where \( x \) is the position along the length of the rod, measured from one end.

### Step 2: Express the Mass Element
The mass element \( dm \) of a small segment \( dx \) of the rod can be expressed as:
\[ dm = \lambda(x) \, dx = (2 + x) \, dx \]

### Step 3: Set Up the Integrals for the Center of Gravity
The position of the center of gravity \( x_{cg} \) can be calculated using the formula:
\[ x_{cg} = \frac{\int_0^L x \, dm}{\int_0^L dm} \]
where \( L \) is the length of the rod (3 m in this case).

### Step 4: Calculate the Denominator (Total Mass)
First, we need to calculate the total mass \( M \) of the rod:
\[ M = \int_0^3 dm = \int_0^3 (2 + x) \, dx \]
Calculating this integral:
\[ M = \int_0^3 (2 + x) \, dx = \left[ 2x + \frac{x^2}{2} \right]_0^3 = \left( 2(3) + \frac{3^2}{2} \right) - \left( 0 \right) \]
\[ = 6 + \frac{9}{2} = 6 + 4.5 = 10.5 \, \text{kg} \]

### Step 5: Calculate the Numerator (Moment about the Origin)
Next, we calculate the moment about the origin:
\[ \int_0^3 x \, dm = \int_0^3 x(2 + x) \, dx = \int_0^3 (2x + x^2) \, dx \]
Calculating this integral:
\[ \int_0^3 (2x + x^2) \, dx = \left[ x^2 + \frac{x^3}{3} \right]_0^3 = \left( 3^2 + \frac{3^3}{3} \right) - \left( 0 \right) \]
\[ = 9 + 9 = 18 \, \text{kg m} \]

### Step 6: Calculate the Center of Gravity
Now we can find the position of the center of gravity:
\[ x_{cg} = \frac{\int_0^3 x \, dm}{M} = \frac{18}{10.5} \]
Calculating this gives:
\[ x_{cg} = \frac{18}{10.5} = \frac{36}{21} = \frac{12}{7} \, \text{m} \]

### Conclusion
The position of the center of gravity of the rod is:
\[ \boxed{\frac{12}{7} \, \text{m}} \]