Download The gravitational force law: Application

PHYS 172: Modern Mechanics
Spring 2011
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Lecture 6: Gravity, Iterative prediction, Electrical force, Uncertainty Read 3.3–3.10
The gravitational force law: Application
m2
Newton
r2 −1
Fgrav on 2 by1 = −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×m 2
kg 2
Gravitational constant
Gravitational force on a planet
Fgrav on 2 by1 = −G
m2 m1
r2−1
2
rˆ2−1
star
planet
r1 = 2,1,1.5 ×1011 m
r2 = 3,3.5, −0.5 ×1011 m
m1 = 4 ×1030 kg
r2−1
m2 = 3 ×1024 kg
1. Calculate r2 −1 ≡ r2 − r1
r2−1 = 1, 2.5, −2 ×1011 m
(1×10
2. Distance r2−1 =
11
on 2 by1
2
2
2
1, 2.5, −2 ×1011 m
r2−1
=
= 0.299, 0.746, −0.597
r2−1
3.35 × 1011 m
3. Unit vector: rˆ2 −1 =
3. Force: Fgrav
m ) + ( 2.5 ×1011 m ) + ( −2 ×1011 m ) = 3.35 ×1011 m
= −G
m2 m1
r2−1
2
rˆ2−1 = −7.16 ×1021 0.299,0.746,-0.597 N
Fgrav on planet bystar = 7.16 × 1021 -0.299,-0.746,0.597 N
magnitude
direction
Gravitational force on a planet
Fgrav on 2 by1 = −G
m2 m1
r2−1
2
rˆ2−1
r2−1
star
planet
m1 = 4 × 1030 kg
m2 = 3 × 1024 kg
r1 = 2,1,1.5 × 1011 m
r2 = 3,3.5, −0.5 × 1011 m
Fgrav on planet bystar = 7.16 × 1021 -0.299,-0.746,0.597 N
Checking results:
1. Diagram
2. Order of magnitude
3. Units
4. Unit vector
Clicker question # 1:
What is the gravitational force exerted by the planet on the star?
A) The same
21
B) Fgrav on star by planet = 7.16 × 10 0.299,0.746,-0.597 N
21
C) Fgrav on star by planet = −7.16 × 10 0.299,0.746,-0.597 N
Gravitational force near the Earth’s surface
m
RE
Fgrav on m by M E = −G
M Em
RE
2
rˆ
~ The same for all objects on surface
Fgrav on m by M E = gm
Gravitational
field
ME = 5.976 1024 kg
RE = 6.37 106 m
g = −G
ME
RE
2
rˆ
The magnitude: g = 9.8 N/kg
Fg = mg
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 2 by1 = −G
p
m2 m1
r2−1
2. Update momentum
2
rˆ2−1
p f = pi + Fnet ∆t
Choose ∆t short enough
(F & v 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 mi =
i≠ j
Fm j on mi
2. Update momentum of each mass
p f = pi + Fnet ∆t
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 ∆t
4. Repeat
∆t is a critical parameter!
Iterative prediction of motion: throwing a rock
Constant Force (mg)
+ Momentum Principle
= Projectile Motion (curved path)
But, can also add air resistance = non-constant force
Example: mass on spring, equilibrium
How far will the spring stretch?
s
∆p ∆t = Fnet
0 = Fnet
Equilibrium:
momentum does not change
0 = 0, ks s − mg , 0
0 = ks s − mg
ks s
mg
s = mg / ks
Example: mass on spring, in motion
ks s
ks s
mg
mg
mg
As spring stretches, force gets larger.
Electric force: the electric charges
Charges: property of an object
• Two types: positive (+) and negative (-)
• Like charges: repel.
• Opposite charges: attract
• Net charge of a system:
algebraic sum of all the charges
• Conservation of charge
• The force exerted by one point charge
on another acts along the line
joining the charges
Charge: measured in C (Coulomb)
Elementary charge: e = 1.602 10-19 C
Charge of electron is –e, of a proton +e
The electric force law (Coulomb’s law)
Felec on1by 2
q1
q2
r2−1
r̂2−1
Coulomb’s law
Felec on 2 by1
Felec on 2by1 =
1
4πε 0
1
q2 q1
4πε 0 r2−1 2
= 9 ×109
rˆ2 −1
N×m 2
C2
Electric force versus gravity
Gravity (Cavendish, 1798)
mm
Fg = G 1 2 2
r
G = 6.67 10-11 m3/(kg.s2)
3m
m1 = m2 = 70kg
Fg = 0.000000036 N
Electric force (Coulomb, 1795)
1 q1 q2
Fe =
4πε 0 r 2
1/(4πε0) = 8.99 109 Nm2/C2
3m
q1 = q2 ≈ 1028 ⋅ e = 1.6 × 109 C
Fe = 2.6 × 1027 N
Predicting the future of a gravitational system
Massive star
And small planets
fixed
Two body: ellipse (or circle)
fixed
Determinism:
If we know the positions and momenta of all particles
in the Universe we can predict the future
Is there free will?
Predicting the future of gravitational system
Solar system
Binary star
Sun, Earth and Moon
Problems:
Sensitivity Initial condition and ∆t
Inability to account for all interactions
! 1025 molecules in glass of water !
Small particles: quantum mechanics
Probability and uncertainty
Example: a free neutron decays with ~15 minutes:
n → p + + e− +ν
Probability
t
Clicker:
Can we predict the motion of an electron near a free neutron?
A)Yes
B) No