Physics 106 Week 9
Newton’s Law of Gravitation
SJ 7th Ed.: Chap 13.1 to 2, 13.4 to 5
• Newton’s inverse-square law of gravitation
ƒ Force
F
ƒ Gravitational acceleration “g”
•
•
•
•
Superposition
Gravitation near the Earth’s surface
Gravitation inside the Earth (concentric shells)
Gravitational potential energy
ƒ Related to the force by integration
ƒ A conservative force means it is path independent
ƒ Escape
E
velocity
l it
Goal
Newton’s Universal Law of Gravitation
1
Newton’s Law of Universal Gravitation
For a pair of point masses
Direction: towards each other
m1
G
F21
3rd law pair of
forces
G
F12
G
r12
Displacement
from m1 to m2
Magnitude:
m2
| F12 | =
G m1m2
Direction:
Forces point to each other
Æ Attractive force
2
r12
Gravitational Constant G is small: G=6.67x10-11 Nm2/kg2
Gravitational force:
Too small to notice between most human-scale objects and smaller
Large if at least one of two objects is massive, like earth-human, earth-sun
Gravitational force “mg” you learned so far:
A special case of universal gravitation law
Gravitational force between earth and objects near the
surface of the earth
Fgravity = mg
| F12 | =
applies only between earth and objects
near the surface of earth
G m1m2
Relation:
applies to any objects
2
r12
g =G
M Earth
REarth 2
2
Example:
(A) Find the gravitational force between two 100 kg
persons separated by 1 m.
(B) Find the gravitational force between Sun and Earth.
Mass of Sun is 2 x 10^30 kg, mass of earth is 6 x 10^24 kg
Distance between sun and earth is 1.5 x 10^11 m.
Gravitational force by multiple point masses:
The net force on a point mass when there are many others nearby is the
vector sum of the forces taken one pair at a time
m2
G
r12
G
Fon 1 =
G
G
G
G
∑ Fi,1 = F2,1 + F3,1 + F4,1
i≠1
All gravitational effects are between
pairs of masses.
G
F21
m1
G
F41
G
F31
G
r13
m3
G
r14
m4
3
iClicker Quiz
m1 = m2 = m3 = m4 = m5 =1kg
m4
m2
Distances from m1 to m2,
m3, m4, m5 are all 1 m.
What is magnitude of the
m1
m5
m3 net gravitational force on
m1 by m2, m3, m4, and
m5?
A)5 G x kg^2/m^2,
kg^2/m^2
B)4 G x kg^2/m^2
C)3 G x kg^2/m^2
D)2 G x kg^2/m^2
E)zero
iClicker Quiz
m1 = m2 = m3 =1kg
Distances from m1 to m2,
and m3 are all 1 m.
What is magnitude of the
m1
m3 net gravitational force on
m1 by m2 and m3?
m2
A)3 G x kg^2/m^2,
kg^2/m^2
B)2 G x kg^2/m^2
C)Sqrt(2) G x kg^2/m^2
D) G x kg^2/m^2
E) zero
4
Gravitational force between a point-like object and
continuous mass distribution with spherical symmetry
Shell Theorem I:
For a mass OUTSIDE of a uniform spherical shell of mass,
the shell’s
shell s gravitational force (field) is the same as that
of a point mass concentrated at the shell’s center
m
r
m
x
r
x
Same for a solid sphere (e.g., Earth, Sun)
m
r
m
x
Shell Theorem:
r
x
superposition for masses with spherical symmetry
Shell Theorem II:
For a test mass INSIDE of a uniform spherical shell of
mass the shell’s gravitational force (field) is zero
mass,
m
x x
• Obvious by symmetry for center
• Elsewhere, proved by integrating over sphere
5
iClicker Q
Which two objects have the greatest gravitational force between them?
Objects are all spherically symmetric or point mass.
Radius =200 km
Mass= 4*10^6 kg
A) Point Mass= 10 kg
Distance from the center = 3000 km
B) Point Mass= 10 kg
Radius =100 km
Mass= 4*10^6 kg
Distance from the center = 3000 km
C) Point Mass= 10 kg
Point Mass= 4*10^6 kg
Distance = 3000 km
D) They all have the same gravitational force.
E) None of the above
Gravitation near the surface of the Earth:
What do “g” and “weight = mg” mean?
m
• Earth’s mass acts as like a point mass Mearth
at the center (by the Shell Theorem)
• Radius of Earth = Rearth
• Object with mass m is at altitude h…
…above the surface, so r = Rearth + h
h
re
mg = G
me
When m is “on or
near the surface:
h << Rearth
g ≅
mM earth
( Rearth + h) 2
or, in other words
G M earth
2
Rearth
where
R earth + h ≈ R earth
R earth =6,370 km
M_earth = 6 x 10^24 kg
More generally, free fall acceleration depends on altitude
gh = G
M earth
( Rearth + h) 2
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Altitude dependence of g
ƒ Weight decreases with
altitude h
Free fall acceleration
9.3 What is the magnitude of the free-fall acceleration at a point that is
a distance 2re above the surface of the Earth, where re is the radius
of the Earth?
a)
b)
c)
d)
e)
4.8
1.1
3.3
2.5
6.5
m/s2
m/s2
m/s2
m/s2
m/s2
ag =
G me
(re + h) 2
at any altitude
g = 9.8 m/s 2
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