A body A of mass M while falling wertically downwards under gravity brakes into two parts, a body B of mass `(1)/(3)` M and a body C of mass `(2)/(3)` M. The center of mass of bodies B and C taken together shifts compared to that of body A towards

Options

A. (a)depends on height of breaking
B. (b)does not shift
C. (c)body C
D. (d)body B

Explanation

To solve the problem, we need to analyze the situation step by step:

### Step 1: Understand the Initial Setup
We have a body A of mass M falling vertically under gravity. When it breaks into two parts, body B has a mass of \( \frac{1}{3}M \) and body C has a mass of \( \frac{2}{3}M \).

**Hint:** Visualize the scenario by drawing a diagram of body A falling and then breaking into two parts.

### Step 2: Identify the Center of Mass of the Initial Body
The center of mass (CM) of body A, which is a single mass M, is located at its geometric center. Since it is falling vertically, we can assume that the CM is moving downwards under the influence of gravity.

**Hint:** Remember that the center of mass of a uniform body is at its center.

### Step 3: Analyze the Center of Mass of Bodies B and C
When body A breaks into bodies B and C, we need to find the new center of mass for the two parts. The center of mass of a system of particles can be calculated using the formula:

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

Where:
- \( m_1 \) and \( m_2 \) are the masses of bodies B and C.
- \( x_1 \) and \( x_2 \) are their respective positions.

**Hint:** Use the masses of bodies B and C to find their center of mass.

### Step 4: Consider the Movement of the Center of Mass
Since body A is falling under gravity, the center of mass of the entire system (A, B, and C) will continue to fall downwards. When body A breaks apart, the center of mass of bodies B and C will also fall, but we need to determine if it shifts compared to the original center of mass of body A.

**Hint:** Think about how the masses of B and C are distributed and how that affects the center of mass.

### Step 5: Determine the Shift in Center of Mass
In this case, since there are no external horizontal forces acting on the system and the bodies are falling vertically, the center of mass of the two parts (B and C) will not shift horizontally. It will remain aligned with the center of mass of body A.

**Hint:** Consider the principle of conservation of momentum and the fact that the center of mass of a closed system remains constant if no external forces act on it.

### Conclusion
The center of mass of bodies B and C taken together does not shift compared to that of body A. It remains in the same vertical line as body A, which means it does not shift towards A, B, C, or D.

**Final Answer:** The center of mass of bodies B and C does not shift compared to that of body A.