A solid cube of the edge a is molten and moulded in eight identical small solid cubes and are placed on one other on a straight line with the edge of the bottom cube on the same horizontal plane on which big cube was placed , then the vertical shift in the centre of mass is

Options

A. `(3a)/2`
B. 2a
C. `(5a)/2`
D. 3a

Answer

To solve the problem of finding the vertical shift in the center of mass when a solid cube of edge \( a \) is melted and molded into eight identical smaller cubes, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Volume of the Original Cube**:
The volume \( V \) of the original cube with edge length \( a \) is given by:
\[ V = a^3 \]

2. **Calculate the Volume of Each Smaller Cube**:
Since the original cube is melted and molded into 8 identical smaller cubes, the volume of each smaller cube \( V' \) is:
\[ V' = \frac{V}{8} = \frac{a^3}{8} \]

3. **Find the Edge Length of Each Smaller Cube**:
Let the edge length of each smaller cube be \( a' \). The volume of a cube is given by the cube of its edge length, so:
\[ V' = (a')^3 \]
Setting the two expressions for volume equal gives:
\[ (a')^3 = \frac{a^3}{8} \]
Taking the cube root of both sides:
\[ a' = \frac{a}{2} \]

4. **Determine the Initial Center of Mass**:
The center of mass of the original cube is at its geometric center, which is at a height of:
\[ h_i = \frac{a}{2} \]

5. **Position of the Smaller Cubes**:
When the 8 smaller cubes are stacked one on top of the other, the total height of the stack is:
\[ \text{Total height} = 8 \times a' = 8 \times \frac{a}{2} = 4a \]
The center of mass of the stack of cubes is located at the height of half the total height of the stack:
\[ h_f = \frac{4a}{2} = 2a \]

6. **Calculate the Vertical Shift in the Center of Mass**:
The vertical shift \( \Delta h \) in the center of mass is given by the difference between the final and initial heights:
\[ \Delta h = h_f - h_i = 2a - \frac{a}{2} \]
Simplifying this expression:
\[ \Delta h = 2a - \frac{a}{2} = \frac{4a}{2} - \frac{a}{2} = \frac{3a}{2} \]

### Final Answer:
The vertical shift in the center of mass is:
\[ \Delta h = \frac{3a}{2} \]