# Linear momentum and CM

1. Jul 8, 2008

### Eric [Tsu]

As an example, if you raise your arms in front of you, your center of mass will move slightly forward. However, you would not fall down because your feet's platform won't allow it. Does this constitute linear momentum, or does the entire system need to move in order to have linear momentum?

In other words, is linear momentum defined as the movement of a systems center of mass or of the entire system? Thanks.

Last edited: Jul 8, 2008
2. Jul 8, 2008

3. Jul 8, 2008

### physicsnoob93

I'm not too sure about this, but i think its the movement of the center of mass. I say this because if you throw a spinning object in a projectile, it moves along the center of its mass. Even when you drop something, and it's spinning, it's center of mass will draw a straight line in perfect conditions.

And since our calculations do not require any "conversion" of the center of mass to the whole linear system, i think that linear momentum is defined as the movement of the system's center of mass.

Heh, i might be wrong though, just what i think.

4. Jul 8, 2008

### Staff: Mentor

Only an hour and 14 minutes and you're complaining?

More seriously,

The linear momentum of a system is defined as the sum of the linear momenta of its parts. It can be proven to be equal to the mass of the system times the velocity of the center of mass:

$$\sum {m_i {\vec v}_i} = M {\vec v}_{cm}$$

where

$$M = \sum {m_i}$$

Last edited: Jul 8, 2008
5. Jul 8, 2008

### Eric [Tsu]

So, would the body in my original example have linear momentum? With the external force preventing the motion being the platform of the feet?

6. Jul 9, 2008

### Staff: Mentor

If you raise your arms in front of you while standing on a frictionless surface, your center of mass will not move since there would be no external force to move it (your feet would slide backwards a bit). Of course the friction of the ground against your feet does provide such a force, which gives you some linear momentum--albeit briefly. Friction in the opposite direction will quickly bring you back to zero momentum.

7. Jul 9, 2008

### Staff: Mentor

For simplicity, assume you move your arms so the center of mass moves only vertically.

While you begin to raise your arms and your center of mass accelerates upward, the upward force exerted by the floor on your feet increases so that the net force on your body (gravity plus floor) is nonzero and upward. This net force is what produces the acceleration of your center of mass.

While you stop raising your arms and your center of mass decelerates to a stop at its final position, the force exerted by the floor on your feet decreases so that the net force on your body is nonzero and downward. This net force is what produces the deceleration of your center of mass.

At other times (before, afterwards, and while the center of mass is moving upward at constant speed) the upward force exerted by the floor has its normal value so the net force is zero.