Question

Consider a system comprising two extended bodies, which have masses $M_1$ and $M_2$ and centers of mass at ${\mathbf{R}}_1$ and ${\mathbf{R}}_2$. Prove that the $\mathrm{CM}$ of the whole system is at $$\mathbf{R}=\frac{M_1{\mathbf{R}}_1+M_2{\mathbf{R}}_2}{M_1+M_2}$$ This beautiful result means that in finding the CM of a complicated system, you can treat its component parts just like point masses positioned at their separate centers of mass - even when the component parts are themselves extended bodies.

Short answer

The CM of the system is at $\mathbf{R}=\frac{M_1{\mathbf{R}}_1+M_2{\mathbf{R}}_2}{M_1+M_2}$.

Step-by-step solution

Understand the Concept of Center of Mass

The examples involving a single point mass suggest that the center of mass (CM) is a location where the total mass of a system seems to be concentrated. In this exercise, we are dealing with two extended bodies, which means each body has its own center of mass at position vectors ${\mathbf{R}}_1$ and ${\mathbf{R}}_2$, respectively.

Breakdown the Mathematical Expression

The formula to find the CM of two point masses, $M_1$ and $M_2$, located at positions ${\mathbf{R}}_1$ and ${\mathbf{R}}_2$ is given as $\mathbf{R}=\frac{M_1{\mathbf{R}}_1+M_2{\mathbf{R}}_2}{M_1+M_2}$. This tells us that the position vector of the CM $\mathbf{R}$ is a weighted average of the position vectors of the two masses.

Apply the Formula to Extended Bodies

When dealing with extended bodies, each body can be treated as if all of its mass is concentrated at its respective center of mass. Thus, the formula still applies to extended bodies, interpreting ${\mathbf{R}}_1$ and ${\mathbf{R}}_2$ as the centers of mass of each extended body.

Derive the Center of Mass for the System

The formula $\mathbf{R}=\frac{M_1{\mathbf{R}}_1+M_2{\mathbf{R}}_2}{M_1+M_2}$ essentially states that the center of mass of the system of two objects is the point at which their masses balance each other out around the origin. By algebraically combining these terms, we determine the location where the entire mass appears to be concentrated.

Verify with Conceptual Understanding

This result embodies a property that simplifies the analysis of a multi-component system, allowing each component to be reduced to a point mass positioned at its center of mass. This means that even if the bodies are complex and extended, their interactions with other parts of the system can be simplified by considering their centers of mass.

Key concepts

Extended Bodies

When we talk about extended bodies, we refer to objects that have size and shape, rather than being mere points. This means they stretch across space and their mass is not concentrated at a single point, unlike point masses.
Imagine holding a large book in your hand. It’s not enough to think of the book’s mass as being in one tiny spot; instead, we look at how the book’s weight is spread out across its entire area.

For extended bodies, each part of the object can have a different position and mass. Yet, these bodies still have a center of mass (CM), which acts as a single point where the body's entire mass can be thought to reside. For a system comprised of multiple extended bodies, each having their own CM, we can treat each body’s CM as if all its mass is gathered there.
This approach simplifies the complex systems because even if an extended body has different parts, we only focus on its center of mass when calculating for systems or interactions involving multiple such bodies.

Weighted Average

When finding the center of mass for two bodies, we use a mathematical tool called a weighted average. Weighted averages allow us to consider not just position but also the importance or contribution of each element—here, the contribution is based on mass.
Think about finding which point best represents two weighted points in space. Each point has a position, and because they have mass, they also have weight in this calculation. When calculating the CM using the formula $\mathbf{R}=\frac{M_1{\mathbf{R}}_1+M_2{\mathbf{R}}_2}{M_1+M_2}$, each position vector is multiplied by its corresponding mass, $M_1$ or $M_2$. This multiplication represents the 'weight' each point contributes to the final result.

• $M_1{\mathbf{R}}_1$ contributes to the result based on how large $M_1$ is.
• Similarly, $M_2{\mathbf{R}}_2$ adds its weight.

The total mass $M_1+M_2$ acts as a normalizing factor, allowing us to find a balance point that illustrates the overall 'average' position, but weighted by their masses.

Two-Body System

A two-body system involves studying the interactions between two separate masses, which might be extended bodies. Each body in the system has its own mass and center of mass, identified by $M_1,{\mathbf{R}}_1$ and $M_2,{\mathbf{R}}_2$, respectively.

In our system, we are interested in where their combined center of mass is located. This is important in both physics and engineering because it helps predict how bodies behave together, such as when they experience forces.
By calculating the center of mass using $\mathbf{R}=\frac{M_1{\mathbf{R}}_1+M_2{\mathbf{R}}_2}{M_1+M_2}$, we learn about equilibrium points and how two spatially separated masses collaborate as one system.

Consider a seesaw balanced perfectly in the middle. The position and weight of each person on either side affect how the seesaw balances. Similarly, the center of mass provides insight into how these two objects will interact as a singular body. Understanding the center of mass helps clarify the results of any forces applied to them, predict movement, and ensure stability within the system.