Transcript F c = mv 2 /r

Chapter 8:
Universal
Gravitation
Chapter 8 Objectives
Relate - Kepler’s laws of planetary motion to
Newton’s Laws of universal gravitation.
Calculate - Periods and speeds of orbiting
objects.
Describe - Cavendish’s method for measuring
the universal gravitation constant (G).
Concept Development Map
Examples/Applications
Falling Objects
What is it?
Attraction, Pull, Fall, Drop,
Move, towards center
of earth
Definitions
Weight, heavy
(gravis)
Satellite Orbit
Weightlessness
Sports
Flight, Ballooning
Bouncing; Bungie
Amusement Rides
Slowing Trucks; objects
Gravity
Grave
(serious;
sober)
Importance,
significance
8.1
Motions in the
Heavens and on Earth
Johannes Kepler (1571-1630)
Johannes Kepler became an assistant to Tycho Brahe
(1546-1601) in Prague. Brahe had very precise data on
planetary motion but believed that the earth was the
center of the universe. Kepler wanted to use a suncentered model of planetary motion along with Brahe’s
data. This led to Kepler’s three laws of planetary
motion.
Kepler’s Three Laws of Planetary Motion
Law 1:
The Paths of the Planets are elliptical, with the
sun at one focus.
Discussion:
Why don’t we feel really cold as an
entire planet when we are far from
the sun?
Kepler’s Three Laws of Planetary Motion
Law 2:
Imaginary line from sun to planet sweeps out
equal areas in equal time, so planets move faster
when close to sun and slower when farther away.
Discussion:
Why not feel accelerations when
we speed up and slow down in
the seasons?
eliptical motion animation
Kepler’s Three Laws of Planetary Motion
Law 3:
Square of the ratio of the periods of any two
planets about the sun is equal to the cube of the
ratio of their average distance from the sun.
TA 2 rA 3
    
TB  rB 

Kepler’s Laws of Planetary Motion
Newton’s Law of Universal Gravitation
In 1666, Isaac Newton knew that a force kept the
planets in orbit. To follow Kepler’s laws the
magnitude of the force must vary inversely with
the square of the distance between their centers
and must be proportional to the masses of the
two planets. This combined into:
mA mB
F G
2
d
The constant of proportionality turned out to be a universal number
(capital G) that holds anywhere in the universe.
Little g versus Big G
mearth = 5.974×1024 kg
mA mB
F G
 mA g
2
d
dearth = rearth = 6,378 km
G = 6.673 × 10-11 m3 kg-1 s-2

5.974(10)24 kg
(6.378(10)3
km)2
6.673(10)-11 m3
kg
s2
(1 km)2
(103
m)2
= 9.8 m/s2 !!!!!
Newton’s Law of Universal
Gravitation
The Inverse Square Law is
Everywhere
Cavendish’s Measurement of “G”
In 1798, Englishman Henry Cavendish (17311810) used an apparatus similar to the one below
to measure the universal gravitation constant G.It
had taken over a hundred years before someone
could measure it accurately. A metal rod was
attached to a ceiling support and the twisting of
the metal caused a measurable force.
Von Jolly: Measuring
“G” More Accurately
Discovering Neptune
8.1 Vocabulary
Gravitational Force: The attractive
force that exist between all objects.
Law of Universal Gravitation:
Gravitational force between any two
objects is directly proportional to the
product of the masses and inversely
proportional to the square of the
distance between their centers.
8.2
Using the Law of
Universal Gravitation
Satellite Velocity
CA Standard 1.f. and 1.g.
g. Students know applying a force to an object
perpendicular to the direction of its motion
causes the object to change direction but not
speed (e.g., Earth's gravitational force causes a
satellite in a circular orbit to change direction but
not speed).
f. Students know circular motion requires the
application of a constant force directed toward
the center of the circle.
Satellite Velocity
Centripetal Force and acceleration.
Centripetal acceleration always points to the
center of the circle or ellipse. Its magnitude is
equal to the square of the speed, divided by the
radius of motion.
ac = v2/r
Centripetal force is mass times the centripetal
acceleration.
Fc = mv2/r
Satellite Velocity
Newton’s Second Law:
Centripetal Acceleration:
F = ma
ac = v2/r
Force Centripetal:
Fc = mv2/r
Satellite in Free Fall
Figure 8.6: Satellites are in a state of constant free fall.
Satellite Velocity
Force Centripetal:
mv
Fc 
r
Newton’s Inverse Square Law:
mA mB
F G
2
d
2
Combining:
2
mv
mA mB
G
2
r
d
Satellite Velocity
Solving for velocity:
2
mS v
mE mS
G 2
r
r
mE
v G
r

2
Or
mE
v G
r
Satellite Velocity
Solving for velocity:
But:
Leads to:

mE
v G
r
d 2r
mE
v

 G
t

r

r3
  2
GmE
Weight and Weightlessness
Weightlessness
“If a person falls freely, he won’t feel his own weight.
This simple thought made a deep impression on me.”
- Albert Einstein.
On that day, the physicist's daydream ended with
what he later called his "happiest moment." He
surmised that the unlucky painter would feel
weightless when accelerating toward the ground.
This clue led Einstein to reason that gravity and
acceleration must be equivalent. Called the
"equivalence principle," this idea was the seed that
- over the next nine years - bloomed into Einstein's
masterpiece, the "General Theory of Relativity." This
new theory laid the foundation for relativistic
astrophysics and modern cosmology.
Apparent Weightlessness

Two Kinds of Mass
Inertial Mass:
Fnet
minertial 
a
Gravitational Mass:
2
Fgrav r Fgrav
mgravitational 

g
GmE
Intertial Mass: Truck accelerates on level ground.
Block of ice slides to back of the truck bed.

Gravitational Mass : Truck goes with constant
velocity up a mountainous slope. Again the block
of ice slides to back of the truck bed. It is pulled
to the earth’s center.
Einstein’s Theory of Gravity
Einstein’s used his General Theory of Relativity to
explain the effect of gravity. The concept of
gravitational field allows us to picture the way gravity
acts on bodies far away. However, neither theory
explains the origin of gravity. According to Einstein,
gravity is an effect on space itself. One way to
picture it is to place round masses on a sheet of
rubber or elastic fabric. The greater the mass, the
more it makes a depression (indentation) in the
fabric. If an small object gets close to the large
depression, then its motion will be deflected.
But don’t forget that this “Plane” can be rotated in any direction!!
8.2 Vocabulary
Gravitational Mass: Ratio of
gravitational force exerted on an
object to its acceleration due to
gravity.
Inertial Mass: Ratio of net force
exerted on an object to its
acceleration. The mass of an object as
measured by its resistance to
acceleration.
7.3 Circular Motion
Circular Motion
Centripetal Force: A force
that acts towards the center
of the circle of motion. From
the Latin centri (center) +
petere (to move toward).
Centrifugal Force: A force that
acts away from the center of
the circle of motion. From the
Latin centri (center) + fugere
(to flee).
Circular Motion
Centripetal Acceleration:
a = v2/r
F = ma = m (v2/r)
Centripetal Force
Rotational Speed
Circular Motion
Torque
Torque (Latin: torquere, to twist): A force that is applied
perpendicular to the lever arm that results in rotational
motion around an axle. A tangential force multiplied by the
radius of the part it rotates.
 = (Fcosl
What angle gives maximum torque?
What angle gives minimum torque?
Torque in Balance: Seesaw
The torque is balanced on both sides of the seesaw. They
are equal and opposite (Counter-clockwise vs. Clockwise ).
 = 
F1 l1= F2 l2
m1g l1= m2g l2