Rotation about the Center of Mass

In summary, the center of mass follows the same path as if there was no rotational energy because objects naturally rotate around the center of mass. This is due to the fact that the center of mass is a point and rotation does not affect its location. This can be seen through the equations of motion for a system of point masses, where the center of mass experiences no acceleration and is therefore a natural choice for the origin of rotational motion.
  • #1
ninevolt
21
0
Why is it that when an object moves both transitionally and rotationally the center of mass follows the path as if there was no rotational energy?

My theory is that objects naturally rotate around the center of mass and since the center of mass is a point, rotation doesn't affect its location. If this is true then I have another question.

Why do objects naturally rotate around the center of mass?
 
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  • #2
Simple example. Consider just two masses, m1 and m2 joined by a weightless rod of length L.

Imagine some forces F1 and F2 are applied to the two masses. In addition, there may be some tension in the rod. It applies force T to m1 and -T to m2. (Newton's 3rd Law)

Acceleration of m1:

[tex]m_1 a_1 = F_1 + T[/tex]

[tex]a_1 = \frac{F_1 + T}{m_1}[/tex]

Acceleration of m2:

[tex]m_2 a_2 = F_1 - T[/tex]

[tex]a_2 = \frac{F_2 - T}{m_2}[/tex]

Lets say that m1 is located at some point r1 and m2 at r2. Then the center of mass, r, is located at:

[tex]r = \frac{m_1 r_1 + m_2 r_2}{m_1 + m_2}[/tex]

From this, it is easy to convince yourself that acceleration of center of mass, a, is given by the following.

[tex]a = \frac{m_1 a_1 + m_2 r_2}{m_1 + m_2}[/tex]

Substituting earlier equations for a1 and a2 into the above, we get this.

[tex]a = \frac{F_1 + T + F_2 - T}{m_1 + m_2}[/tex]

Tensions cancel. If I now call the total mass of the system M = m1 + m2, the result is exactly what you expect.

[tex]a = \frac{F_1 + F_2}{M}[/tex]

[tex]a M = F_1 + F_2[/tex]

Or in other words, the Newton's 2nd Law is valid for center of mass motion.

It is easy enough to extend the above for any collection of point masses. If you know a bit of calculus, you can derive it for continuous rigid bodies.

As far as the object rotating "around center of mass", any kind of motion can be separated into linear and rotational, with center of rotation being an arbitrary point. The above equations show that in absence of an external force, center of mass moves with no acceleration. That is, uniformly, along a straight line. Therefore, it is a natural choice for origin of rotational motion. You can pick any other point, but that point is going to experience a net force from its neighbors, and therefore, its motion is much more difficult to describe.
 

1. What is rotation about the center of mass?

Rotation about the center of mass refers to the movement of an object around its own center of mass. This is the point at which an object's mass is evenly distributed in all directions, making it the most stable point for rotation.

2. Why is rotation about the center of mass important?

Rotation about the center of mass is important because it allows for stable and controlled movement of objects. It also helps in understanding the dynamics of rotating systems and can be used to analyze and predict the behavior of objects in motion.

3. How is the center of mass determined for an object?

The center of mass for an object can be determined by finding the average position of all the mass of the object. This can be calculated using the formula: x̄ = (m1x1 + m2x2 + ... + mnxn) / (m1 + m2 + ... + mn), where x̄ is the center of mass, mx is the mass at a specific position, and n is the number of mass elements in the object.

4. What is the difference between rotation about the center of mass and rotation about a fixed point?

The main difference between rotation about the center of mass and rotation about a fixed point is the stability of the rotation. When an object rotates about its center of mass, it remains stable and balanced. On the other hand, rotation about a fixed point can cause the object to wobble or tip over if the center of mass is not directly above the fixed point.

5. How does the distribution of mass affect rotation about the center of mass?

The distribution of mass plays a crucial role in rotation about the center of mass. Objects with a more even distribution of mass will have a more stable and predictable rotation, while objects with an uneven distribution of mass may experience more wobbling or tipping during rotation. This is why the center of mass is an important concept in understanding the dynamics of rotating objects.

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