Why does center of mass behave

[xailer]

Why when we throw an extended object like a baseball bat will the center of mass follow the same parabolic path that we expect for a smaller obeject like ball, while the bat itself will rotate around this ceneter of mass?

Why does center of mass behave like a simpler object?

thank you

 

[Doc Al]

If you imagine that an extended body (mass = M) as a set of smaller mass elements (mass = m_1, m_2,...), Newton's 2nd law implies that $F_{net} = M a_{cm}$.

It goes like this. Start with the definition of center of mass:
$$M\vec{r}_{cm} = m_1 \vec{r}_1 + m_2 \vec{r}_2 \ ...$$

Now take the second derivative:
$$M\vec{a}_{cm} = m_1 \vec{a}_1 + m_2 \vec{a}_2 \ ...$$

From Newton's 2nd law applied to each mass element, recognize that:
$$M\vec{a}_{cm} = \vec{F}_1 + \vec{F}_2 \ ...$$
$$M\vec{a}_{cm} = \vec{F}_{net} = \vec{F}_{external}$$

Note that internal forces cancel out and only external forces count. This can be summarized by saying that the cm of an object (or collection of particles) moves as though all the mass were concentrated at the cm and all the external forces were applied at that point.

(For more details, consult any first year physics text.)

 

[xailer]

What does $a_{cm}$ mean? Acceleration of a part of an object that is 1 cm in length?

If that is the case, then $$M\vec{r}_{cm}$$ must mean torque of a part of the body that is 1 cm in length?

$$M\vec{r}_{cm} = m_1 \vec{r}_1 + m_2 \vec{r}_2 \ ...$$

Isn't that a formula for finding how far from the rotation axis a center of mass is? Aren't rotational and translational movements two different things. I don't see how you can combine the two if that is what you are doing here

Note that internal forces cancel out and only external forces count.

what would internal forces be in this case?

This can be summarized by saying that the cm of an object (or collection of particles) moves as though all the mass were concentrated at the cm and all the external forces were applied at that point.

I'm sorry but I can't seem to be able to see the connection

(For more details, consult any first year physics text.)

I did and they don't answer my questions

 

[quantumdude]

What does $a_{cm}$ mean? Acceleration of a part of an object that is 1 cm in length?

If that is the case, then $$M\vec{r}_{cm}$$ must mean torque of a part of the body that is 1 cm in length?

No, the subscript "cm" stands for "center of mass".

Isn't that a formula for finding how far from the rotation axis a center of mass is? Aren't rotational and translational movements two different things. I don't see how you can combine the two if that is what you are doing here

No, that's the formula for locating the center of mass.

what would internal forces be in this case?

It could be anything that results in zero net force: Gravitational attraction, a string that connects the two masses, whatever.