PHYSICS 231 INTRODUCTORY PHYSICS I Lecture 11

PHYSICS 231
INTRODUCTORY PHYSICS I
Lecture 11

Last Lecture
•
Angular velocity, acceleration
  f   i


t
t f  ti
•
•

 f  i
t
Rotational/ Linear analogy 
s  r
vt  r

at  r
  x
 0  v0
 f  vf
 a
t t
(angle in radians)
2

v
2
• Centripetal acceleration: a
(to center)


r

cent

r

Newton’s Law of
Universal
Gravitation
m1m2
FG 2
r
3 

m
11
G  6.67  10 
 kg  s 2 
• Always attractive
• Proportional to both
masses
• Inversely
proportional to
separation squared
Gravitation Constant
• Determined experimentally
• Henry Cavendish, 1798
• Light beam / mirror
amplify motion
Weight
• Force of gravity on Earth
GME m
Fg 
RE2
• But we know


Fg  mg
GME
 g 2
RE
Example 7.14
Often people say astronauts feel
weightless, because there is no
gravity in space.
This explanation is wrong!
What is the acceleration due to
gravity at the height of the space
shuttle (~350 km above the earth
surface)?
8.81 m/s2
(0.90 g)
Example 7.14 (continued)
Correct explanation of
weightlessness:
• Everything (shuttle, people,
bathroom scale, etc.) also falls with
same acceleration
• No counteracting force (earth’s
surface)
• “Accelerating Reference Frame”
• Same effect would be felt in
falling elevator
Example 7.15a
Astronaut Bob stands atop the highest mountain of
planet Earth, which has radius R.
Astronaut Ted whizzes around in a circular orbit at
Carol
the same radius.
Astronaut Carol whizzes around in a circular orbit of
radius 3R.
Astronaut Alice is simply falling straight downward
and is at a radius R, but hasn’t hit the ground yet.
Which astronauts experience
weightlessness?
A.) All 4
B.) Ted and Carol
C.) Ted, Carol and Alice
Ted
Bob
Alice
Example 7.15b
Astronaut Bob stands atop the highest mountain of
planet Earth, which has radius R.
Astronaut Ted whizzes around in a circular orbit at
Carol
the same radius.
Astronaut Carol whizzes around in a circular orbit of
radius 3R.
Astronaut Alice is simply falling straight downward
and is at a radius R, but hasn’t hit the ground yet.
Assume each astronaut weighs
Bob
w=180 lbs on Earth.
The gravitational force acting on
Ted is
A.) w
B.) ZERO
Ted
Alice
Example 7.15c
Astronaut Bob stands atop the highest mountain of
planet Earth, which has radius R.
Astronaut Ted whizzes around in a circular orbit at
Carol
the same radius.
Astronaut Carol whizzes around in a circular orbit of
radius 3R.
Astronaut Alice is simply falling straight downward
and is at a radius R, but hasn’t hit the ground yet.
Assume each astronaut weighs
Bob
w=180 lbs on Earth.
The gravitational force acting on
Alice is
A.) w
B.) ZERO
Ted
Alice
Example 7.15d
Astronaut Bob stands atop the highest mountain of
planet Earth, which has radius R.
Astronaut Ted whizzes around in a circular orbit at
the same radius.
Carol
Astronaut Carol whizzes around in a circular orbit of
radius 3R.
Astronaut Alice is simply falling straight downward
and is at a radius R, but hasn’t hit the ground yet.
Assume each astronaut weighs
w=180 lbs on Earth.
Bob
The gravitational force acting on
Carol is
A.) w
B.) w/3
Ted
C.) w/9
D.) ZERO
Alice
Example 7.15e
Astronaut Bob stands atop the highest mountain of
planet Earth, which has radius R.
Astronaut Ted whizzes around in a circular orbit at
the same radius.
Carol
Astronaut Carol whizzes around in a circular orbit of
radius 3R.
Astronaut Alice is simply falling straight downward
and is at a radius R, but hasn’t hit the ground yet.
Which astronaut(s) undergo an
acceleration g=9.8 m/s2?
A.)
B.)
C.)
D.)
E.)
Alice
Bob and Alice
Alice and Ted
Bob, Ted and Alice Ted
All four
Bob
Alice
Kepler’s Laws
• Tycho Brahe (1546-1601)
• Extremely accurate
astronomical observations
• Johannes Kepler (1571-1630)
• Worked for Brahe
• Used Brahe’s data to find
mathematical description of
planetary motion
• Isaac Newton (1642-1727)
• Used his laws of motion and
gravitation to derive Kepler’s
laws
Kepler’s Laws
1)
2)
3)
Planets move in elliptical orbits with Sun at
one of the focal points.
Line drawn from Sun to planet sweeps out
equal areas in equal times.
The square of the orbital period of any planet
is proportional to cube of the average
distance from the Sun to the planet.
Kepler’s First Law
• Planets move in
elliptical orbits with
the Sun at one focus.
• Any object bound to
another by an inverse
square law will move
in an elliptical path
• Second focus is
empty
Kepler’s Second Law
• Line drawn from Sun to
planet will sweep out equal
areas in equal times
• Area from A to B equals
Area from C to D.
True for any central force due to
angular momentum conservation (next chapter)

Kepler’s Third Law
• The square of the orbital period of any planet is
proportional to cube of the average distance from the
Sun to the planet.
R3
 Constant
2
T
• The constant depends on Sun’s mass, but is
independent of the mass of the planet
Derivation of Kepler’s Third Law
Fgrav
GMm
2

 macent  m R
2
R
2

T
R 3 GM
 2 
2
T
4
m
M
Example 7.16
Data: Radius of Earth’s orbit = 1.0 A.U.
Period of Jupiter’s orbit = 11.9 years
Period of Earth’s orbit = 1.0 years
Find: Radius of Jupiter’s orbit
5.2 A.U.
Example 7.17
Given: The mass of Jupiter is 1.73x1027 kg
and Period of Io’s orbit is 17 days
Find: Radius of Io’s orbit
r = 1.85x109 m
Gravitational Potential Energy
• PE = mgh valid only near
Earth’s surface
• For arbitrary altitude
Mm
PE  G
r
• Zero reference level is
at r=
Example 7.18
You wish to hurl a projectile from the surface of the
Earth (Re= 6.38x106 m) to an altitude of 20x106 m
above the surface of the Earth. Ignore rotation of the
Earth and air resistance.
a) What initial velocity is required?
a) 9,736 m/s
b) What velocity would be required in order for the
projectile to reach infinitely high? I.e., what is the
escape velocity?
b) 11,181 m/s
c) (skip) How does the escape velocity compare to the
velocity required for a low earth orbit?
c) 7,906 m/s
Chapter 8
Rotational Equilibrium
and
Rotational Dynamics
Wrench Demo
Torque
• Torque, t , is tendency of a force to
rotate object about some axis
t  Fd
• F is the force
• d is the lever arm (or moment arm)
• Units are Newton-meters
Door Demo
Torque is vector quantity
• Direction determined by axis of twist
• Perpendicular to both r and F
• Clockwise torques point into paper.
Defined as negative
• Counter-clockwise torques point out of paper.
Defined as positive
r
-
r
F
+
F
Non-perpendicular forces
t  Fr sin 
Φ is the angle between F and r
Torque and Equilibrium
• Forces sum to zero (no linear motion)
Fx  0 and Fy  0
• Torques sum to zero
(no rotation)
t  0
Meter Stick Demo
Axis of Rotation
• Torques require point of reference
• Point can be anywhere
• Use same point for all torques
• Pick the point to make problem least difficult
(eliminate unwanted Forces from equation)
Example 8.1
Given M = 120 kg.
Neglect the mass of the beam.
a) Find the tension
in the cable
b) What is the force
between the beam and
the wall
a) T=824 N
b) f=353 N
Another Example
Given: W=50 N, L=0.35 m,
x=0.03 m
Find the tension in the muscle
W
x
L
F = 583 N