The position of the centre of mass of a cube of uniform mass density will be at

Options

A. the centre of one face
B. the centre of the interaction of diagonals of one face.
C. the geometric centre of the cube
D. the edge of a cube

Explanation

To determine the position of the center of mass of a cube with uniform mass density, we can follow these steps:

### Step 1: Understand the Concept of Center of Mass
The center of mass of an object is the point at which the mass of the object can be considered to be concentrated. For symmetrical objects with uniform mass density, the center of mass is located at the geometric center.

### Step 2: Identify the Geometry of the Cube
A cube has six faces, twelve edges, and eight vertices. The geometric center of a cube can be found by locating the midpoint of the cube's dimensions.

### Step 3: Calculate the Geometric Center
For a cube with side length 'a':
- The coordinates of the vertices can be taken as (0,0,0), (a,0,0), (0,a,0), (0,0,a), (a,a,0), (a,0,a), (0,a,a), and (a,a,a).
- The geometric center (or centroid) can be calculated as the average of the coordinates of the vertices.

### Step 4: Find the Coordinates of the Center of Mass
The coordinates of the center of mass (CM) can be calculated as follows:
- CM_x = (0 + a + 0 + 0 + a + a + 0 + a) / 8 = a/2
- CM_y = (0 + 0 + a + 0 + a + 0 + a + a) / 8 = a/2
- CM_z = (0 + 0 + 0 + a + 0 + a + a + a) / 8 = a/2

Thus, the center of mass of the cube is located at the point (a/2, a/2, a/2), which is the geometric center of the cube.

### Step 5: Conclusion
Since the center of mass of a cube with uniform mass density is located at its geometric center, the correct answer is:
**The geometric center of the cube.**