Download Chapter 6: Uniform Circular Motion and Gravity

Chapter 6:
Uniform Circular
Motion and Gravity
Brent Royuk
Phys-111
Concordia University
Angular Measure
•  Angular distance: Δθ = θ - θo
–  Analogous to linear distance
–  Rotation instead of translation
–  How far did you run? 33 m. How far did you turn?
33o
•  Radians
–  All thetas are measured in radians
–  There are 2π radians in a circle
–  A radian is dimensionless
•  180° = πR
•  mR = (m° π)/180°
–  Importance
•  The radian equation: s = rθ
–  Example: If you run 32 m on a circular track with r
= 200 m, what is your Δθ?
Angular Speed & Velocity
•  Definition: ω =
– 
– 
– 
– 
Δθ
Δt
θ = ωt
Average vs. instantaneous
Example: Make half a turn in 2.0 s: ω = ?
MKS
€ Units
•  Other units
–  What is the angular speed for a 45 rpm record?
•  How far would above object turn in 85 s? How many
revolutions is this?
•  Angular Velocity
– 
– 
– 
– 
Velocity implies?
Two options only
Right-hand rule
also used: (+), (–), cw, ccw
•  Tangential Velocity
–  vt = rω
Angular & Tangential Speeds
•  How fast would a bug have to run to
stay motionless on a record?
•  vt = r ω
•  What is tangential speed of a point at
the edge of an 8.0 cm disk spinning at
ω = 3.4 s-1?
•  Other measurable quantities: Period
(T), frequency (f, ν)
Circular Motion
• 
Uniform Circular Motion
– 
• 
• 
What does centripetal mean?
Derivation
– 
• 
What is the nature of the speed, velocity and
acceleration?
Small-angle approximation
What is a centrifugal force?
ac =
€
€
v2
r
FC =
mv2
r
Circular Motion
• 
Swing a mass on a string, calculate and
measure the centripetal force.
A 1200-kg car rounds a corner of radius 45
m. If the coefficient of friction between the
tires and road is 0.82, what is the greatest
speed the car can have in the corner without
skidding?
• 
– 
• 
What if the roadway is banked inward 18o (no
friction)?
What does “centrifugal” mean?
– 
A “fictitious force.”
The Law
• 
Newton’s Law of
Universal Gravitation
• 
What’s universal about
Newton’s Universal Law
of Gravitation?
m1m2
The Law: F = G
r2
• 
€
Gravity
• 
• 
• 
G = 6.67 x 10-11 N m2/kg2
Superposition
Gravitational Attraction of Spherical Bodies
– 
• 
Why don’t we ever notice attractions between two
terrestrial bodies?
Physical attraction on a love seat.
• 
– 
• 
“Let each person be a uniform-density sphere…”
Big G, Little g
– 
• 
The Shell Theorem
gEverest = 9.78 m/s2
Gravity is an Inverse Square Law
– 
What if you halve the distance between objects?
• 
• 
Double it?
The silly-looking platform illustration.
Cavendish’s Torsion Balance
Weighing the Earth
Huh?
Gravity
• 
gshuttle = 9.08 m/s2, 7.44% less than Earth’s
surface.
Gravity and Orbit
Why are astronauts weightless in orbit?
Kepler’s Laws
•  The puzzle of retrograde motion
Kepler’s Laws
•  The puzzle of retrograde motion
•  The geocentric solution: Epicycles
Kepler’s Laws
•  Tyco Brahe (1546-1601)
–  Brahe’s observatory: 1582, the
Island of Hveen, near Copenhagen
–  Non-telescopic measurements with
a precision of one arc-minute
–  Brahe’s unusual solar-system model
–  The Tychonian Society
Kepler’s Laws
•  Johannes Kepler (1571-1630)
–  “By the study of the orbit of Mars, we must either
arrive at the secrets of astronomy or forever remain
in ignorance of them.”
•  First two laws: 1609
•  Third law: 1619
Kepler’s House in
Regensburg,
Germany
Kepler’s First Law
•  Planets follow elliptical orbits, with the
Sun at one focus of the ellipse
Kepler’s Second Law
•  As a planet moves in its orbit, it sweeps
out an equal amount of area in an equal
amount of time.
–  Why? Conservation of L.
–  There’s no torque about the Sun.
Kepler’s Second Law
Kepler’s Third Law
•  The period, T, of a
planet increases as its
mean distance from
the Sun, r, raised to
the 3/2 power. That
is,
T α r 3/2
€
Actually,
4 π 2mr
T
2
=G
T =
mM S
r
2π
2
GM
, so
r 3 /2
Kepler’s Esoterica
Kepler’s Esoterica
Orbits
•  What is the orbital speed of a satellite orbiting
100 km above the Earth?
•  Find the radius of an orbit above the equator
such that the satellite completes one orbit in
one day.
–  Geosynchronous Orbit
–  Compare with the space shuttle, which orbits at an
altitude of about 150 km.
–  What is the space shuttle’s orbital period?
•  GPS satellites orbit at an average altitude of
12,400 miles = 20,000 km.
–  Which travels faster, a GPS satellite or a
geosynchronous satellite?
Geosynchronous/GPS
Orbital Maneuvers
•  The Hohmann transfer
Black Holes
Tides
Retrograde Motion Explained