Two point objects of mass 1.5 g and 2.5 g respectively are at a distance of 16 cm apart, the centre of mass is at a distance x from the object of mass 1.5 g. Find the value of x.

Options

A. 9 cm
B. 8 cm
C. 10 cm
D. 11 cm

Explanation

To find the distance \( x \) from the object of mass \( 1.5 \, \text{g} \) to the center of mass of the two objects, we can use the formula for the center of mass (CM) of a system of particles. The center of mass \( x_{cm} \) for two point masses is given by:

\[ x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]

where:
- \( m_1 \) and \( m_2 \) are the masses of the two objects,
- \( x_1 \) and \( x_2 \) are their respective positions.

### Step 1: Define the masses and positions
Let:
- \( m_1 = 1.5 \, \text{g} \) (mass of the first object)
- \( m_2 = 2.5 \, \text{g} \) (mass of the second object)

We can place the first mass at the origin (0 cm):
- \( x_1 = 0 \, \text{cm} \)

The second mass is located 16 cm away from the first mass:
- \( x_2 = 16 \, \text{cm} \)

### Step 2: Substitute values into the center of mass formula
Now we can substitute the values into the center of mass formula:

\[ x_{cm} = \frac{(1.5 \, \text{g} \cdot 0 \, \text{cm}) + (2.5 \, \text{g} \cdot 16 \, \text{cm})}{1.5 \, \text{g} + 2.5 \, \text{g}} \]

### Step 3: Calculate the numerator and denominator
Calculate the numerator:

\[ = 0 + (2.5 \cdot 16) = 40 \, \text{g cm} \]

Calculate the denominator:

\[ = 1.5 + 2.5 = 4.0 \, \text{g} \]

### Step 4: Calculate the center of mass
Now we can calculate \( x_{cm} \):

\[ x_{cm} = \frac{40 \, \text{g cm}}{4.0 \, \text{g}} = 10 \, \text{cm} \]

### Step 5: Determine the distance \( x \)
Since \( x \) is the distance from the mass \( 1.5 \, \text{g} \) to the center of mass, we find that:

\[ x = x_{cm} = 10 \, \text{cm} \]

### Final Answer
Thus, the value of \( x \) is \( 10 \, \text{cm} \).

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