Center of Mass

Newtons 2nd Law and Momentum

1. Forces cause changes in an object’s momentum. That is, forces cause the object’s velocity to change over time.

2. We can determine the change in an object’s velocity over time using this expression in the form:

of

if

tt

pp

t

p

F

A special point…

•If the net external force on a system of particles is zero, then (even if the velocity of individual objects changes), there is a point associated with the distribution of objects that moves with zero acceleration (constant velocity).

•This point is called the “center of mass” of the system. It is the balancing point for the mass distribution.

Location of Center of Mass

The CM could be located:• within the object (human

standing straight)

• outside the object (high jumper as she goes over the bar)

Center of Mass is outside the object.

Center of Mass of:

• System of Particles

• Extended Object

Center of Mass of two Objects in one Dimension

• mi is the mass of each object

• xi is the position of each objects center of mass with respect to the origin

Example 1: Center of Mass in one Dimension

• Find the CM of a system of two spheres that have a mass of 2 kg each. One is located 3 cm from the origin on the + x-axis and one is 4 cm from the origin on the – x-axis

• Answer: -.5 cm

Center of Mass of a Rod• Find the center of mass of a rod of mass M and length L.

Ans: xCM = L / 2

CENTER OF MASS (CONT.)•The overall translational motion of a system of particles can be analyzed using Newton's laws as if all the mass were concentrated at the center of mass and the total external force were applied at that point.

Internal and External Forces

• There are two classes of forces that act on and within systems.

• INTERNAL FORCES are forces between an object within the system and another object within the system.

• EXTERNAL FORCES are forces between an object within the system and an object outside the system.

Only External Forces Change the Total

Momentum of the System• Recall Newton’s third law. For every force there is an equal

and opposite force. These “paired” forces are called Newton’s third law pairs.

• Two forces are a Newton’s third law pair if

1. The two different forces are between the same two objects

2. The object exerting the force and the object being acted on switch roles in the two forces.

• Third Law force pairs cancel out. So only forces external to a system matter. ALL internal forces are third law pairs.

Horse and the Cart

Horse and the Cart

• If there are no external forces this is a true statement. Not only will the center of mass not be able to start moving, but if it is, there can be no way for it to stop if it is moving.

• The only way for the center of mass of the system can possibly change its momentum is that there must be a net EXTERNAL force acting on it.

• It is still possible for individual parts to change their momentum (by preforming an explosion), but the center of mass cannot no change momentum.