Center of Mass and Momentum Lecture 08 Monday: 9 February 2004

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.

CENTER OF MASS  M = m 1 + m 2 + ··· M =  m n

Position, Velocity and Acceleration of CENTER OF MASS

MOMENTUM For a system of particles, total momentum is: P = p 1 + p 2 + p 3 ··· or P =  m n v n along with P = Mv cm

MOMENTUM

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.

MOMENTUM For a particle, translational momentum is: Momentum is a vector Its direction is the same as the object’s velocity Momenta must be added as vectors

MOMENTUM (Review) For a particle, translational momentum is: Momentum is a vector Its direction is the same as the object’s velocity Momenta must be added as vectors

MOMENTUM of a SYSTEM (Review) A system is any group of objects that we wish to consider as a group. For a system of particles, momentum is: This is a vector sum!! We have to take direction into account.

What does imply? (Review) 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:

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.

Conservation of Momentum

states that the momentum does not change over time. That is, it is constant. If you know the momentum at any one time, then you know it for all other times.

EXAMPLE: F Ext = 0, F Int  0 Two masses, initially at rest. P Before = P After 0 = m 1 v 1 + m 2 v 2 m 1 v 1 = – m 2 v 2 If, m 1 = m 2 v 1 = – v 2

Relationship between Force and Momentum

What does imply? 1.Forces cause changes in an object’s momentum over time. For constant mass objects, that means 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:

Reworking

IMPULSE In many cases, the integral is easily evaluated as the area under a curve. Time (sec) F (N)

MOMENTUM of a SYSTEM A system is any group of objects that we wish to consider as a group. For a system of particles, the total momentum is: This is a vector sum!! We have to take direction into account.

MOMENTUM (Review) For a particle, translational momentum is: Momentum is a vector Its direction is the same as the object’s velocity Momenta must be added as vectors

MOMENTUM of a SYSTEM (Review) A system is any group of objects that we wish to consider as a group. For a system of particles, momentum is: This is a vector sum!! We have to take direction into account.

What does imply? (Review) 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:

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.

Conservation of Momentum

states that the momentum does not change over time. That is, it is constant. If you know the momentum at any one time, then you know it for all other times.

Conversation of Momentum This does not mean that the momentum of any one object in the system stays the same. It means that if you add up all of the momenta for all of the objects in the system that this total doesn’t change as time passes. This is only true for systems on which no net external force acts.

When motion occurs in two dimensions, TWO conservation of momentum equations are required. Momentum is conserved or not conserved in each direction SEPERATELY.

When motion occurs in two or more directions, we need to consider each direction separately..

Conversation of Momentum in Two Dimensions

Remember, you know how to get the x and y components of velocity….. V V x =VSin  V y = VCos  

In systems made up of several objects, momentum can be transferred from one object to another and still be conserved. However, momentum in the x direction can not be transferred into the y direction and visa-versa. Momentum Transfers

Collisions Momentum is conserved only if there is no net external force acting on the system. Remember, if a system is composed of two or more objects and these objects undergo a collision, then all forces between the colliding objects are INTERNAL forces and so sum to zero. There can be external forces acting on a system in which a collision occurs. An example is friction between the objects and the surface on which they move.

EXAMPLE: F Ext = 0, F Int  0 Two masses, initially at rest. P Before = P After 0 = m 1 v 1 + m 2 v 2 m 1 v 1 = – m 2 v 2 If, m 1 = m 2 v 1 = – v 2