Physics 2011
Lecture 5:
Gravitation and Applying Newton's
Laws
Chapter 12: Gravitation
• Gravity: Action at a distance
Gravitation
(According to Newton, Anyway)
• Newton determined that amoon / g = 0.000278
• and noticed that RE2 / R2 = 0.000273
amoon
g
R
RE
• This inspired him to propose the
Universal Law of Gravitation:
where G = 6.67 x 10 -11 m3 kg-1 s-2
|FMm |= GMm / R2
Gravity...
• The magnitude of the gravitational force F12
exerted on an object having mass m1 by
another object having mass m2 a distance R12
away is:
F12  G
m1 m2
2
R12
m1
F12
F21
m2
R12
• The direction of F12 is attractive, and lies along
the line connecting the centers of the masses.
Gravity...
• Near the Earth’s surface:
– R12 = RE
• Won’t change much if we stay near the Earth's
surface.
• i.e. since RE >> h, RE + h ~ RE.
h
M Em
Fg  G 2
RE
m
Fg
M
RE
Gravity...
• Near the Earth’s surface...
Fg  G
• So |Fg| = mg = ma
–
a=g
Where: g  G
 ME 
ME m
 G 2 

m
2
RE
 RE 

=g
All objects accelerate with
acceleration g, regardless of
their mass!
ME
2

9
.
81
m
/
s
RE2
Example gravity problem:
• What is the force of gravity exerted by the earth on a
typical physics student?
– Typical student mass m = 55kg
– g = 9.8 m/s2.
– Fg = mg = (55 kg)x(9.8 m/s2 )
– Fg = 539 N
Fg
• The force that gravity exerts on any
object is called its Weight
W = 539 N
Force and acceleration
• Suppose you are standing on a bathroom scale on
Earth and it says that your weight is W. What will the
same scale say your weight is on the surface of the
mysterious Planet X ?
• You are told that RX ~ 20 REarth and MX ~ 300 MEarth.
(a)
0.75 W
(b)
1.5 W
(c)
2.25 W
E
X
Solution
• The gravitational force on a person
of mass m by another object (for instance
Mm
F

G
a planet) having mass M is given by:
2

R
W X FX
Ratio of weights = ratio of forces:

WE FE
MX m
R X2

M m
G E2
RE
G
M
 X
ME
2
WX
1
 300    .75
 20 
WE
R 
 E 
 RX 
2
Newton’s Third Law:
• Forces occur in pairs: FA ,B = - FB ,A.
– For every “action” there is an equal and opposite
“reaction”.
• This is consistent with the discussion of
gravitation:
m1
m2
F12
F21
R12
F12  G
m1 m2
 F21
2
R12
Newton's Third Law...
• FA ,B = - FB ,A. is true for contact forces as
well:
Fm,w
Fw,m
Ff,m
Fm,f
Particles in Equilibrium (2-D)
• A particle is in Equilibrium when the sum of all
forces on the body is Zero (FNET = 0)
• By Superposition, Equilibrium can be computed
for each dimensional component separately
( ΣFx = 0, Σ Fy = 0)
Dynamics in 2-D
• When a body is not in Equilibrium, a Net
Force exists in at least one dimensional
component: ΣFi = Finet = mai
(Note that the Normal Force, n, is perpendicular to the
15 degree Plane)
Friction
• The friction force is proportional to the
Normal Force: f = μN
Static vs. Kinetic Friction
• As the names imply, Static friction is
present only when a body is not moving
with respect to the surface it is contacting.
• Kinetic friction is moving friction.
• Static friction, representing the force
required to ‘break free’ and start
movement is generally greater than its
Kinetic counterpart.
fs > fk
Friction in Fluids
• At low speeds, f = kV
• At high speeds, f = DV2
Dynamics of Angular Motion
• In angular motion, the Net Force is in the
same direction as the Centripetal
Acceleration
In UCM, Centripetal Acceleration is
the only game at the Fair: