∆ r
∆ t
=
Momentum
( f
− v i
∆ t
)
= r net alternative statement of
Newton’s second law
The time rate of change of momentum of an object is equal to the net force acting on it p t

„
I r
When a single, constant force F acts on the object for time ∆ t, there is an impulse
I r delivered to the object:
= r
F
∆ t = area under F-t curve
F
I r
Using
= r
F
∆ r
=
∆ r
=
∆ t m ( v r f
− v r i
)
=
Impulse r
F
∆ t t

„
Momentum is conserved for the system of objects. You must define the isolated system
„
Assumes only internal forces are acting during the collision
„
Can be generalized to any number of objects

„
„
Momentum is conserved in any collision
Inelastic collisions
„
Kinetic energy is not conserved
„
Some of the kinetic energy is converted into other types of energy such as heat, sound, work to permanently deform an object
„
Perfectly inelastic collisions occur when the objects stick together
„
Not all of the KE is necessarily lost

• The average angular speed, ω , of a rotating rigid object is the ratio of the angular displacement to the time interval
 av

 f
  t f
 t i i 
 
 t
Unit: rad/s

object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change:
 av

 f
  t f
 t i i 
 
 t
Unit: rad/s 2

• Displacements
 s
   r
• Speeds v t
  r
• Every point on the rotating object has the same angular motion
• Not every point on the rotating object has the same linear motion
• Accelerations a t
  r

T

2



2  v r f

1

T

2 
Time to go around once v

Centripetal Acceleration
– The magnitude of the centripetal acceleration is given by a c
 v t
2 r
– This direction is toward the center of the circle


The angular velocity and the linear velocity are related ( v t
= ω r )
The centripetal acceleration can also be related to the angular velocity a c
  2 r

• General equation F c
 m a c
 m v
2 r
• If the force vanishes, the object will move in a straight line tangent to the circle of motion

of the acceleration is due to changing speed a t
 a c
 a
 a t a to changing direction c
• Total acceleration can be found from these components a
 a t
2  a
2
C
Don’t confuse: a t
  r a c v
2
  r

2 r

Newton’s Law of Universal Gravitation
• Every particle in Universe attracts every other particle with force: directly proportional to product of masses inversely proportional to square of distance between them.
F

G m
1 m
2
2 r inverse square law
G
= universal gravitational constant
= 6.673 x 10 -11 N m² /kg²

Applications of Universal Gravitation
Gravitational force of uniform sphere on particle outside sphere same as force exerted if entire mass of the sphere concentrated at its center--Gauss’ Law
• Acceleration due to gravity
• g will vary with altitude g

G
M r
2
E

Gravitational Potential Energy
• PE = mgy is valid only near the earth’s surface
• For objects high above the earth’s surface, an alternate expression is needed
PE
 
G
M
E m r
– Zero reference level is infinitely far from the earth

Escape Speed
• speed needed for an object to “escape” from planet
• For the earth, v esc
11.2 km/s is about
• Note, v is independent of the mass of the object v esc

2 GM
E
R
E

Ch. 8 General Torque Formula
Component of F
 r
OR  r F sin

• The lever arm , d , is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force d = r sin
 so
 = F d = F r sin 

Torque and Equilibrium
• First Condition of Equilibrium
 


F x
0
 or
0 and 

F y
 0
•The Second Condition of Equilibrium states
0

• If an object is “symmetrical’’ then the center of mass is just where you think that it should be.
• If an object is “complicated” then
 r

 m

M

R
CoM
The torque from gravity is “as if” the force of gravity acts only at the CoM

Torque and Angular Acceleration
Newton’s Second Law for a Rotating Object
 I  analogous to
F = ma
I = moment of inertia
For Uniform Ring
I   m r 2  MR 2 moment of inertia depends on quantity of matter and its distribution and location of axis of rotation

Other Moments of Inertia