PHYS 172: Modern Mechanics
Spring 2011
Lecture 3 – Relativity, The Momentum Principle
Read 2.1-2.4
Average rate of change of momentum
The stronger the interaction, the faster is the change in the momentum
Average rate of change of momentum:
∆p p f − pi
=
∆t
t f − ti
Units:
Instantaneous rate of change of momentum:
dp
∆p
= lim
dt ∆t →0 ∆t
kg ⋅ m
s2
The principle of relativity
Physical laws work in the same way for observers
in uniform motion as for observer at rest
Inertial reference frame
Inertial frame: a frame in which Newton’s first law of motion is valid
Physical laws work in the same way in any inertial frame
? Is the Earth’s surface an inertial reference frame?
Special theory of relativity
Published in 1905.
Based on the principle of relativity and
on experimental fact that c=const in all inertial reference frames
Time dilation: time runs slower in moving reference frames
Length contraction: object length becomes shorter in moving reference frame
Light photon frame: time does not exist, the universe has zero length!
RELATIVITY
“Physical laws work in the same way for observers
in uniform motion as for observers at rest.”
(=in all inertial reference frames)
vavg
The position update formula
rf = ri + vavg ( t f − ti )
RELATIVITY
“Physical laws work in the same way for observers
in uniform motion as for observers at rest.”
(=in all inertial reference frames)
vavg
vavg
The position update formula
rf = ri + vavg ( t f − ti )
Note: all parameters must be measured in respect to the selected reference
frame to predict motion in respect to that reference frame
The Momentum Principle (Newton’s second law)
An object moves in a straight line and at constant speed
except to the extent that it interacts with other objects
The Momentum Principle
∆p = Fnet ∆t
What is “force” F?
p = γ mv
γ=
1
v
1−
c
Change of momentum is equal to the net force
acting on an object times the duration of the
interaction
• Assume that F does not change during ∆t
F units: N ≡ m ⋅ kg/s 2
• measure of interaction.
• defined by the momentum principle.
(Newton)
2
Derivative form of the Momentum Principle
The Momentum Principle
∆p
= Fnet
∆t
∆p = Fnet ∆t
Works only if force is
constant during ∆t
The rate of the momentum change is equal to force
If force changes introduce instantaneous rate of change:
dp
∆p
≡ lim
∆
t
→
0
dt
∆t
The momentum principle
dp
= Fnet
dt
Newton’s Second Law: nonrelativistic form
Newton’s original formulation:
The rate of change of amount of body’s motion is proportional to force
momentum
dp
= Fnet
dt
Assume nonrelativistic case:
Momentum principle is the second Newton’s law
p = mv
d ( mv )
= Fnet
dt
m
dv
(Assume m = const)
= Fnet
dt ≡ a (definition of acceleration)
Newton’s second law
Fnet = ma
Traditional form of 2nd Newton’s law
The principle of superposition
Fpush
Fearth
∆p = Fnet ∆t
Net force
The Superposition Principle:
The net force on an object is the vector sum of all the
individual forces exerted on it by all other objects
Fgravity
Each individual interaction is unaffected by the
presence of other interacting objects
Ignored
friction!
Definition of net force:
Fnet = F1 + F2 + ...
Misconception: need constant force to maintain motion
? Why planets move ?
Clicker question 1
A planet is attracted to two nearby stars as shown.
Which of the red vectors below represents the net force on the planet?
F1
A)
Fnet
Star 1
F2
F1
F1
Fnet
B)
F2
Planet
F1
Fnet
C)
F2
F2
Star 2
Measuring force
The Momentum Principle
1. Use the momentum principle – not convenient
∆p = Fnet ∆t
2. Using Hookes spring force law
L0
Fspring = k S s
L
s = ∆L = L − L0
Robert Hooke
1635-1702
kS – spring stiffness (spring constant)
units: N/m
Direction of force: toward equilibrium
Impulse
The Momentum Principle
∆p = Fnet ∆t
Definition of impulse
Impulse ≡ Fnet ∆t
Note: small ∆t
Fnet ~ const
Momentum principle:
The change of the momentum of an object is equal to the net impulse applied to it
Predictions using the Momentum Principle
The Momentum Principle
∆p = Fnet ∆t
Update form of the momentum principle
p f − pi = Fnet ∆t
p f = pi + Fnet ∆t
p fx , p fy , p fz = pix , piy , piz + Fnet , x , Fnet , y , Fnet , z ∆t
For components:
Short enough,
F~const
p fx = pix + Fnet , x ∆t
p fy = piy + Fnet , y ∆t
p fz = piz + Fnet , z ∆t
Example
.
kS=500 N m
n
rictio
no f
∆p = Fnet ∆t
p f = pi + Fnet ∆t
Force: provided by a spring stretched by ∆L=4 cm
interaction duration: 1 s
? Find momentum pf if pi=<0,0,0> kg.m/s
1. Force:
Fspring = k S ∆L
Fspring = 500(N/m)0.04(m)=20 N
NB: force must not
change during ∆t
Fspring = 20,0,0 N
2. Momentum:
p f = pi + Fnet ∆t
p f =< 0, 0, 0 > kg×m/s + 20, 0, 0 N ⋅ (1 s )
p f =< 20, 0, 0 > kg×m/s
N.s = kg.m/s2.s
= kg.m/s
System and surroundings
p
system
System: an object for which we calculate momentum (car)
a system can consist of several objects
Surroundings: objects which interact with system (earth, man, air…)
p f = pi + Fnet ∆t
! Only external forces matter !
Internal forces cancel
animated slide
Applying the Momentum Principle to a system:
predicting motion
1. Choose a system and surroundings
2. Make a list of objects in surroundings that exert significant forces on system
3. Apply the Momentum Principle
p f = pi + Fnet ∆t
4. Apply the position update formula if needed
5. Check for reasonableness (units, etc.)
rf = ri + vavg ∆t