Download Slide 1

Gravity
• Galileo’s observations on gravity led to
Newton’s Law of Gravitation and the three
Laws of Motion
• Objects fall at the same rate regardless of
mass because more massive objects have
more inertia or resistance to motion
•
Fgrav = G (m1 x m2) / r2
• Force of gravity between two masses is
proportional to the product of masses divided
by distance squared  ‘inverse square law’
Newton – Three Laws of Motion
1. Inertia
2. F = ma
3. Action = Reaction
Newton’s Laws of Motion
• Law of Inertia: A body continues in state of
rest or motion unless acted on by an external
force; Mass is a measure of inertia
• Law of Acceleration: For a given mass m, the
acceleration is proportional to the force applied
F=ma
• Law of Action equals Reaction: For every
action there is an equal and opposite reaction;
momemtum (mass x velocity) is conserved
Velocity, Speed, Acceleration
• Velocity implies both speed and direction;
speed may be constant but direction could
be changing, and hence accelerating
• Acceleration implies change in speed or
direction or both
• For example, stone on a string being whirled
around at constant speed; direction is
constantly changing therefore requires force
Ball Swung around on a String:
Same Speed,
(in uniform circular motion)
Changing Direction
(swinging around the circle)
Ball Swung around on a String:
Same Speed,
(in uniform circular motion)
Changing Direction
(swinging around the circle)
Donut Swung around on a String
Acceleration
Force
Donut Swung around on a String
Acceleration
Force
Conservation of momemtum:
action equal reaction
• The momemtum (mv) is conserved before and
after an event
• Rocket and ignited gases:
M(rocket) x V(rocket) = m(gases) x v(gases)
• Two billiard balls:
m1 v1 + m2 v2 = m1 v1’ + m2 v2’
v1,v2 – velocities before collision
v1’,v2’ – velocities after collision
• Example – you and your friend (twice as
heavy) on ice!
Conservation of momemtum:
action equal reaction
• The momemtum (mv) is conserved before and
after an event
• Rocket and ignited gases:
M(rocket) x V(rocket) = m(gases) x v(gases)
• Two billiard balls:
m1 v1 + m2 v2 = m1 v1’ + m2 v2’
v1,v2 – velocities before collision
v1’,v2’ – velocities after collision
• Example – you and your friend (twice as
heavy) on ice!
Action = Reaction
Equal and Opposite
Force from the Table
Net Force is Zero,
No Net Motion
Force = (apple’s mass)  (acceleration due to gravity)
Action = Reaction
Equal and Opposite
Force from the Table
Net Force is Zero,
No Net Motion
Force = (apple’s mass)  (acceleration due to gravity)
Acceleration due to gravity
• Acceleration is rate of change of velocity, speed or
direction of motion, with time  a = v/t
• Acceleration due to Earth’s gravity : a  g
g = 9.8 m per second per second, or 32 ft/sec2
•
Speed in free-fall
T (sec)
v (m/sec)
v (ft/sec)
0
0
0
1
9.8
32
2
19.6
64
3
29.4
96
60 mi/hr = 88 ft/sec (between 2 and 3 seconds)
Galileo’s experiment revisited
• What is your weight and mass ?
• Weight W is the force of gravity acting
on a mass m causing acceleration g
• Using F = m a, and the Law of Gravitation
W = m g = G (m MEarth) /R2
(R – Radius of the Earth)
The mass m of the falling object cancels
out and does not matter; therefore all
objects fall at the same rate or acceleration
g = GM / R2
i.e. constant acceleration due to gravity 9.8
m/sec2
Galileo’s experiment on gravity
• Galileo surmised that time differences
between freely falling objects may be too
small for human eye to discern
• Therefore he used inclined planes to slow
down the acceleration due to gravity and
monitor the time more accurately
v
Changing the angle of the incline changes the velocity v
‘g’ on the Moon
g(Moon) = G M(Moon) / R(Moon)2
G = 6.67 x 10-11 newton-meter2/kg2
M(Moon) = 7.349 x 1022 Kg
R(Moon) = 1738 Km
g (Moon) = 1.62 m/sec/sec
About 1/6 of g(Earth); objects on the
Moon fall at a rate six times slower than on
the Earth
Escape Velocity and Energy
• To escape earth’s gravity an object must have
(kinetic) energy equal to the gravitational
(potential) energy of the earth
• Kinetic energy due to motion
K.E. = ½ m v2
• Potential energy due to position and force
P.E. = G m M(Earth) / R
(note the similarity with the Law of Gravitation)
• Minimum energy needed for escape: K.E. = P.E.
½ m v2 = G m M / R
Note that the mass m cancels out, and
• v (esc) = 11 km/sec = 7 mi/sec = 25000 mi/hr
The escape velocity is the same for all objects of
mass m
Escape Velocity and Energy
• To escape earth’s gravity an object must have
(kinetic) energy equal to the gravitational
(potential) energy of the earth
• Kinetic energy due to motion
K.E. = ½ m v2
• Potential energy due to position and force
P.E. = G m M(Earth) / R
(note the similarity with the Law of Gravitation)
• Minimum energy needed for escape: K.E. = P.E.
½ m v2 = G m M / R
Note that the mass m cancels out, and
• v (esc) = 11 km/sec = 7 mi/sec = 25000 mi/hr
The escape velocity is the same for all objects of
mass m
Object in orbit  Continuous fall !
Object falls towards the earth at the same rate as the earth curves away from it
Angular Momentum
Conservation of angular momentum says that
product of radius r and momentum mv must be
constant  radius times rotation rate (number of
rotations per second) is constant
Angular Momentum
• All rotating objects have angular momentum
• L = mvr ; acts perpendicular to the plane of
rotation
• Examples: helicopter rotor, ice skater, spinning
top or wheel (experiment)
• Gyroscope (to stabilize spacecrafts) is basically
a spinning wheel whose axis maintains its
direction; slow precession like the Earth’s axis
along the Circle of Precession
Conservation of Angular Momentum
• Very important in physical phenomena
observed in daily life as well as throughout
the Universe. For example,
• Varying speeds of planets in elliptical
orbits around a star
• Jets of extremely high velocity particles,
as matter spirals into an accretion disc and
falls into a black hole
1
Relativistic
Jet “From” Black
Hole
1. “Relativistic velocities are close to the speed of light
Quiz 1
• Each quiz sheet has a different 5-digit
symmetric number which must be filled in
(as shown on the transparency, but NOT
the same one!!!!!)
• Please hand in both the exam and the
answer sheets with your name on both
• Question/answer sheets will be handed
back on Wednesday after class
• Please remain seated until we begin
collecting (20-25 minutes after start)
• Class after quiz
Stars and Galaxies: Galileo to HST
•
http://thenextdigit.com/16961/nasa-telescopes-new-panoramic-view-andromeda-resolves-stars/
Why is the sky blue ?
The atmosphere scatters the blue light more than red light
Light and Matter
• Light is electromagnetic energy, due to
interaction of electrical charges
• Matter is made of atoms – equal number
of positive and negative particles
• An atom is the smallest particle of an
element; natural element H to U
• Atom  Nucleus (protons + neutrons),
with ‘orbiting’ electrons
• No. of protons in nucleus = Atomic
Number
• Science of light  Spectroscopy
Radiation and Spectroscopy
•
•
•
•
Light is electromagnetic energy
Propagates as both particles and waves
Photons – particles of light
Wavelength = Velocity / Frequency
Light is electromagnetic wave;
Does not require a medium to propagate,
unlike water or sound
Wavelength is the distance between successive crests or troughs
WAVES: Frequency, Wavelength, Speed
Wavelength ()
Frequency (f)
(# waves/second)
Speed (c)
Frequency ‘f’ is the number of waves passing a point per second
Speed = wavelength x frequency

c=f
Units of wavelength and frequency
• Frequency is the number of cycles per second
• Since speed of light is constant, higher the
frequency the shorter the wavelength and viceversa
• Wavelengths are measured in Angstroms:
1A = 1/100,000,000 cm = 1/10 nanometer (nm)
• The higher the frequency the more energetic
the wave
• Wavelength (or frequency) defines radiation or
color
Prisms disperse light into its
component colors: Red-Violet
White
Light
Spectrum
Prism
Visible Light
• Forms a narrow band within the
electromagnetic spectrum ranging from
gamma rays to radio waves
• Human eye is most sensitive to which
color?
• Yellow. Why?
Light: Electromagnetic Spectrum
From Gamma Rays to Radio Waves
Gamma
X-Ray
UV
Visible
Gamma rays are the most energetic (highest frequency, shortest wavelength),
Radio waves are the least energetic.
Decreasing
Wavelength
OR
Increasing
Frequency
Visible light spectrum: Each color is defined
by its wavelength, frequency or energy
Red - Blue  7000 - 4000 Angstroms
( 1 nm = 10 A, 1 A = 10-8 cm)
Blue light is more energetic than red light
Light also behaves like ‘particles’ called photons
Photon energy, frequency, wavelength: E = h f = hc/
Planck’s Law (‘h’ is a number known as Planck’s
constant)
Matter and Particles of Light: Quantum Theory
• Light (energy) and matter in motion behave both as
•
•
•
•
•
•
waves and particles
Wave-Particle Duality - Quantum Theory
Particles of light are called photons: E = hf = hc/
Photons of a specific wavelength  may be absorbed
or emitted by atoms in matter
Matter is made of different natural elements: lightest
Hydrogen (1 proton), heaviest Uranium (92 protons)
Smallest particle of an element is atom, made up of a
nucleus (protons and neutrons), and orbiting electrons
Electrons and protons attract as opposite electrical
charges, NOT gravitationally like planets and Sun
The Hydrogen Atom
Electron orbits
Discrete energies
Absorption of light (energy) photon by H-atom
Emission of light photon by H-atom
photon energy color
Series of spectral lines of Hydrogen
Wavelengths of series of lines from Hydrogen
SPECTRAL SIGNATURE OF ELEMENTS
Continuous, Absorption, and Emission Spectra
Brightness and Temperature
• Brightness is related to the total energy
emitted, or the luminosity of an object
• The energy emitted is related to the
temperature of the object
•
B = s T4 (s is a constant)
Stefan-Boltzmann Law
Color Indicates Temperature and
Energy of the Source
Blackbody: Perfect
absorber and emitter
Of radiation at a given
Temperature T
Surface T (Sun) = 5600 K
“ (Mercury) = 800 K
Objects generally emit radiation at all wavelengths, but mostly at one peak
Wavelength depending on their temperature (e.g. blue – hot, red – cool)
TEMPERATURE SCALES
Astronomers usually use the Kelvin Scale
Room Temp = 300 K = 27 C = 81 F
K = C + 273
C = (F - 32) x 5/9
F = (C x 9/5) + 32
~ (F - 30) / 2
~ C x 2 + 30
The Doppler Effect
• Why does the “pitch” of a police siren differ
when, say, a police car is approaching you, or
when you are running away from the police
(not recommended) ?
• The frequency (the number of sound waves
per second) is higher when approaching, and
smaller when receding from the source
Doppler Effect in Sound
High Pitch
(short waves)
Low Pitch
(long waves)
Brightness decreases inversely as the square of the distance
d=1
B=1
d=2
B=1/4
d=3
B=1/9
The Doppler Effect
Velocity c = frequency (f) x wavelength ()
Doppler Shift of Wavelengths
•
•
•
•
•
What about the wavelength?
What about light?
Shorter wavelength  Blue-shift,
Longer wavelength  Red-shift
We can determine the velocity of
astronomical objects, moving away or
towards the Earth, by measuring the
wavelength of light from the object
• Observed red-shift of galaxies all over the sky
shows that galaxies are moving away from one
another  the Universe is expanding
(Hubble’s Law)