A point object of mass m is kept at (a, 0) along x-axis. What mass should be kept at (-3a,0), so that centre of mass lies at origin ?

Options

A. m
B. 2m
C. `(m)/(3)`
D. 3m

Answer

To find the mass that should be placed at (-3a, 0) so that the center of mass of the system lies at the origin, we can follow these steps:

### Step 1: Understand the System
We have two masses:
- Mass \( m \) located at \( (a, 0) \)
- Mass \( M \) located at \( (-3a, 0) \)

### Step 2: Write the Formula for Center of Mass (CM)
The formula for the center of mass \( x_{cm} \) of a system of particles along the x-axis 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,
- \( x_1 \) and \( x_2 \) are their respective positions.

### Step 3: Set the Condition for Center of Mass at the Origin
We want the center of mass to be at the origin, so we set \( x_{cm} = 0 \):

\[ 0 = \frac{m \cdot a + M \cdot (-3a)}{m + M} \]

### Step 4: Simplify the Equation
Now, multiply both sides by \( (m + M) \) to eliminate the denominator:

\[ 0 = m \cdot a - 3M \cdot a \]

### Step 5: Factor Out Common Terms
Since \( a \) is not zero, we can factor it out:

\[ 0 = a(m - 3M) \]

### Step 6: Solve for M
For the equation to hold true, we need:

\[ m - 3M = 0 \]

This leads to:

\[ m = 3M \]

### Step 7: Rearranging the Equation
Now, we can express \( M \) in terms of \( m \):

\[ M = \frac{m}{3} \]

### Conclusion
Thus, the mass \( M \) that should be placed at \( (-3a, 0) \) so that the center of mass lies at the origin is:

\[ M = \frac{m}{3} \]

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