Two balls A and B of masses 100 gm and 250 gm respectively are connected by a stretched spring of negligible mass and placed on a smooth table. When the balls are released simultaneously the initial acceleration of B is 10 cm/sec2 west ward. What is the magnitude and direction of initial acceleration of the ball A -

Options

A. `25cm//sec^(2)` East ward
B. `25 cm//sec^(2)` North ward
C. `25cm//sec^(2)` West ward
D. `25cm//sec^(2)` South ward

Answer

To solve the problem step by step, we will use the concept of the center of mass and the relationship between the accelerations of the two connected masses.

### Step 1: Understand the system
We have two balls, A and B, with masses:
- Mass of A (m1) = 100 gm = 0.1 kg
- Mass of B (m2) = 250 gm = 0.25 kg

They are connected by a spring and placed on a smooth table. When released, ball B has an initial acceleration of 10 cm/s² westward.

### Step 2: Define the acceleration of the center of mass
The acceleration of the center of mass (a_cm) of a system of particles is given by the formula:
\[ a_{cm} = \frac{m_1 a_1 + m_2 a_2}{m_1 + m_2} \]
where \(a_1\) and \(a_2\) are the accelerations of masses A and B respectively.

### Step 3: Set up the equation
Given:
- \(a_2 = -10 \, \text{cm/s}^2\) (westward is taken as negative)
- \(a_{cm} = 0\) (since there are no external forces acting on the system)

Substituting the values into the equation for \(a_{cm}\):
\[ 0 = \frac{(0.1 \, \text{kg}) a_1 + (0.25 \, \text{kg})(-10 \, \text{cm/s}^2)}{0.1 \, \text{kg} + 0.25 \, \text{kg}} \]

### Step 4: Simplify the equation
The total mass \(m_1 + m_2 = 0.1 + 0.25 = 0.35 \, \text{kg}\).

Now, substituting into the equation:
\[ 0 = \frac{(0.1) a_1 - 2.5}{0.35} \]

### Step 5: Solve for \(a_1\)
Multiplying both sides by 0.35 gives:
\[ 0 = 0.1 a_1 - 2.5 \]
Rearranging gives:
\[ 0.1 a_1 = 2.5 \]
Thus,
\[ a_1 = \frac{2.5}{0.1} = 25 \, \text{cm/s}^2 \]

### Step 6: Determine the direction
Since \(a_2\) was negative (westward), and the system's center of mass must remain stationary, the acceleration of ball A must be in the opposite direction (eastward).

### Final Answer
The magnitude of the initial acceleration of ball A is **25 cm/s²** and the direction is **eastward**.

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