AP Physics
Chapter 9 Summary
Section 9.2: Center of Mass
It is important to remember that in physics we often treat mass as a point mass, i.e. we treat an object as if all of its mass is located at one point. Unless otherwise specified, we treat objects as a uniform body and this point is the geometric center of the object.
The Center of Mass for a system of particles is given by the equation: x cm
 m
1
1
 m
2
( m
1 x
1
 m
2 x
2
) where m represents the mass of each body and x represents its position with respect to a fixed point.
Similar equations are used for the velocity and acceleration of the Center of Mass v c
 m
1
1
 m
2
( m
1 v
1
 m
2 v
2
) a c
 m
1
1
 m
2
( m
1 a
1
 m
2 a
2
)
The equations above can be expanded to include many particles by simply adding additional terms for each object.
When analyzing a many particle system it is wise to remember that since we can calculate the center of mass of a system and its velocity and acceleration, we can then analyze the system using the following rule: 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.
A corollary which follows is: If the net external force on a system of particles is zero, then the center of
mass of the system move with a constant velocity.
The Center of Mass for Solid Objects
If an object has spherical symmetry – the center of mass is at the geometric center
If an object is not symmetrical then you calculate the center of mass along each axis and this determines the center of mass of that object for translational motion in that direction.
Section 9.4: Linear Momentum of a Particle
Linear Momentum is the product of the mass of an object and its linear velocity.
ρ = mv where ρ (roe) is the momentum, m is the mass in kilograms, and v is velocity in m/s the resulting unit is a N•s.
The change in Momentum, also called the impulse is equal to the resultant force acting on the body and is the direction of that force. Since ρ = mv and F = ma and a = Δv/t Δρ =mΔv = FΔt
Hence change in momentum is a product of the force acting on an object and the time during which it acts.
Section 9.7: Conservation of Linear Momentum

Like energy – momentum in a system is also conserved. For example, prior to firing a shotgun, neither the gun, nor the shell inside are moving, hence the momentum of the system is zero. m g v g
+ m s v s
= 0
After the gun is fired, the shell moves forward, and by conservation of momentum, the gun moves backward m g v g
= -(m s v s
)
Momentum is always conserved.
Impulse and Momentum
Impulse is the change in momentum, which is a result of the net force acting on a particle during a given time. This is summarized mathematically as FΔt = Δρ.
Section 9.8: Momentum and Kinetic Energy in Collisions
There are two types of Collisions: Elastic and Inelastic.
In an elastic collision:

Two objects collide and there are two objects move away

In a perfectly elastic collision kinetic energy is conserved.
In an inelastic collision:

Two objects collide and one object moves away or one object breaks apart to form two objects.

Kinetic energy is not conserved
Section 9.9: Inelastic Collisions
Inelastic collisions can be describe in one of two ways:

Two objects collide and stick together- hence they move off as one object after the collision

The converse of an inelastic collision is when one object breaks into two pieces and both move off after the breaking (often an explosion)
It is important to remember that while total energy is conserved in the collision, often it is changed into
heat energy or energy of deformation so kinetic energy is not conserved.
Inelastic Collisions can be described mathematically by the equation: m
1 v
1
+ m
2 v
2
= (m
1
+ m
2
)v’
Section 9.10: Elastic Collisions
Elastic Collisions can be described mathematically by the following:

Conservation of Momentum: m
1 v
1
+ m
2 v
2
= m
1 v
1
’ + m
2 v
2
’

Conservation of kinetic energy: v
1
- v
2
= v
2
’ - v
1
’
Section 9.11: Collisions in two dimensions
When analyzing a collision in two dimensions, the key is to remember that momentum is conserved in each direction. ρ ix
= ρ fx
and ρ iy
= ρ fy.
Remembering that since ρ = mv and v is a vector.