what is the C.M. of circle having four equal mass m kg. at any point on circumfrence.

To determine the center of mass (C.M.) of a circle with four equal masses placed at any point on its circumference, we need to consider how the distribution of these masses affects the overall center of mass of the system. Let's break this down step by step.

Understanding the Setup

Imagine a circle with a radius \( R \). We have four identical masses, each of mass \( m \), positioned at equal intervals along the circumference. For simplicity, let's place these masses at the following coordinates on a Cartesian plane:

• Mass 1 at \( (R, 0) \)
• Mass 2 at \( (0, R) \)
• Mass 3 at \( (-R, 0) \)
• Mass 4 at \( (0, -R) \)

Calculating the Center of Mass

The center of mass (C.M.) of a system of particles can be calculated using the formula:

C.M. = \(\frac{\sum m_i x_i}{\sum m_i}, \frac{\sum m_i y_i}{\sum m_i}\)

In our case, since all the masses are equal, we can simplify the calculations:

Step 1: Sum of the x-coordinates

Calculating the x-coordinates:

• Mass 1 contributes \( R \)
• Mass 2 contributes \( 0 \)
• Mass 3 contributes \( -R \)
• Mass 4 contributes \( 0 \)

The total sum of the x-coordinates is:

Sum of x = \( R + 0 - R + 0 = 0 \)

Step 2: Sum of the y-coordinates

Now, calculating the y-coordinates:

• Mass 1 contributes \( 0 \)
• Mass 2 contributes \( R \)
• Mass 3 contributes \( 0 \)
• Mass 4 contributes \( -R \)

The total sum of the y-coordinates is:

Sum of y = \( 0 + R + 0 - R = 0 \)

Final Calculation of the Center of Mass

Now, substituting these sums into the center of mass formula:

C.M. = \(\frac{0}{4m}, \frac{0}{4m} = (0, 0)\)

Conclusion

The center of mass of the system, with four equal masses placed at equal intervals on the circumference of a circle, is located at the center of the circle, which is the origin point \( (0, 0) \). This result makes intuitive sense because the symmetrical arrangement of the masses around the center balances them out perfectly.

In summary, regardless of the specific positions of the masses along the circumference, as long as they are symmetrically distributed, the center of mass will always be at the center of the circle. This principle is a fundamental aspect of physics, illustrating how symmetry plays a crucial role in determining the center of mass in various systems.