A square frame of mass m and side 2a is kept in contact with a circular frame of radius a and mass m. Find distance of center of mass from center of circular frame

To find the distance of the center of mass from the center of the circular frame when a square frame is in contact with it, we can use the concept of the center of mass for composite systems. Let's break this down step by step.

Understanding the Components

We have two objects: a square frame and a circular frame. The square frame has a mass \( m \) and a side length of \( 2a \), while the circular frame has a radius \( a \) and the same mass \( m \). The center of the circular frame is our reference point for calculating the center of mass of the combined system.

Position of the Centers

First, we need to determine the positions of the centers of both frames:

• The center of the circular frame is at the origin, which we can denote as \( (0, 0) \).
• The square frame, being in contact with the circular frame, will have its center located at a distance of \( a \) from the center of the circular frame. Since the square frame has a side length of \( 2a \), its center will be at \( (a, 0) \) if we assume it is positioned to the right of the circular frame.

Calculating the Center of Mass

The center of mass \( (x_{cm}, y_{cm}) \) of a system of particles (or in this case, frames) can be calculated using the formula:

\( x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \)
\( y_{cm} = \frac{m_1 y_1 + m_2 y_2}{m_1 + m_2} \)

Here, \( m_1 \) and \( m_2 \) are the masses of the circular and square frames, respectively, and \( (x_1, y_1) \) and \( (x_2, y_2) \) are their respective positions.

Substituting the Values

In our case:

• For the circular frame: \( m_1 = m \), \( (x_1, y_1) = (0, 0) \)
• For the square frame: \( m_2 = m \), \( (x_2, y_2) = (a, 0) \)

Now substituting these values into the center of mass equations:

\( x_{cm} = \frac{m \cdot 0 + m \cdot a}{m + m} = \frac{ma}{2m} = \frac{a}{2} \)
\( y_{cm} = \frac{m \cdot 0 + m \cdot 0}{m + m} = 0 \)

Final Result

The center of mass of the system is located at \( \left( \frac{a}{2}, 0 \right) \). To find the distance from the center of the circular frame, we simply take the x-coordinate of the center of mass, which is \( \frac{a}{2} \).

Thus, the distance of the center of mass from the center of the circular frame is \( \frac{a}{2} \).