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PHYS 172: Modern Mechanics
Lecture 4 – Physical Models, Fundamental Interactions
Fall 2011
Read 2.7–2.8, 3.1-3.4
Predictions using the Momentum Principle
The Momentum Principle
Dp = Fnet Dt
Update form of the momentum principle
p f - pi = Fnet Dt
p f = pi + Fnet Dt
p fx , p fy , p fz  pix , piy , piz  Fnet , x , Fnet , y , Fnet , z t
For components:
p fx  pix  Fnet , x t
p fy  piy  Fnet , y t
p fz  piz  Fnet , z t
Short enough,
F~const
Example
Dp = Fnet Dt
p f = pi + Fnet Dt
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 = kS DL
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 Dt
( )
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
Physical models
“Spherical cow”
Ideal model: ignore factors that have no significant effect on the outcome
Example: colliding students
Two students are late for class and run into each other head-on.
Q: Estimate the force that one student exerts on the other during collision
Simplest model:
F
Ffloor , N
Fair
Ffloor , P
FEarth
System: one spherical student
Surroundings: earth, floor, air, second spherical student
Force: Earth, floor, air, other student – unknown!
Example: colliding students
y
Ffloor , N
F
Strategy:
p f  pi  Fnet t
p   mv
Fair
Ffloor , P
FEarth
rf  ri  vavg t
x
p f  pi  Fnet t
 0, 0    pix , 0  Ffloor , P  Fair  F , Ffloor , N  FEarth  t
  pix ,0   F ,0  t
pix  F t
Example: colliding students
y
Ffloor , N
F
Strategy:
p f  pi  Fnet t
p   mv
Fair
Ffloor , P
FEarth
pix  F t
What is the collision time?
vavg
x

t
x
t 
vavg
What is the initial momentum?
Find F:
rf  ri  vavg t
x
Assume: vi =5 m/s, x=0.05m
x
t 
 vi  v f  / 2
Assume: m=60 kg
pix 300 kg  m/s
F

 15000 N
t
0.02 s
t  0.02 s
pix  mvix  300 kg  m/s
Newton’s Great Insight:
The force that attracts things toward the earth (e.g. a falling
apple) is the same force that keeps planets orbiting about
the sun
The gravitational force law
m2
Newton
r21
Fgrav on 2by1  G
r̂21
m1
r2
m2
r21  r2  r1
r1
m1
m2 m1
r21
2
rˆ2 1
Cavendish
G  6.7 10
11
N×m2
kg 2
Gravitational constant
Predicting motion of a planet
Where will the planet be after one month?
Use position update formula:
p
rf  ri  vavg t
If we assume that velocity is constant
F
Does not work because the force is
changing the velocity!
The force changes with position.
The momentum changes with position.
In general, there is no algebraic equation to
predict motion of more than 2 interacting
objects.
Iterative prediction of a motion of one planet
Simple case: one planet
star is fixed in space
1. Calculate gravitational force:
Fgrav on 2by1  G
p
m2 m1
r21
2. Update momentum
2
rˆ2 1
p f  pi  Fnet t
Choose t short enough
(F & p do not change much)
F
3. Calculate v and update position
rf  ri  vavg t
4. Repeat
Critical parameter: t
Iterative prediction of motion
Real case: many objects
objects are free to move
1. Calculate net force on each mass:
Fon m = å Fm
i
i¹ j
j
on mi
2. Update momentum of each mass
p f = pi + Fnet Dt
Choose t short enough
(F & v do not change much)
3. Calculate v and update position of each mass
Iterative approach: works for
any kind of force, not just
gravity!
rf = ri + vavg Dt
4. Repeat
t is a critical parameter!