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The Universe
Revision
Gravity and mass
What’s the difference between mass and weight?
• Mass is how much matter (material) is in
an object and is measured in kg.
• Weight is the force of gravity acting on
an object (of known mass) and is
measured in N kg-1.
Newton’s Inverse Square Law
of Gravitation
This law states that there is a force of attraction between
any two objects in the universe.
The size of the force is proportional to the product of the
masses of the two objects, and inversely proportional to the
square of the distance between them.
Gm1m2
F
2
r
m1 and m2 are the masses of
the two objects, and r is the
distance between them.
G = gravitational constant =
6.67 x 10-11 Nm2kg-2
Example 1
• What is the force of • F = Gm1m2
attraction between
r2
two pupils of average
= 6.67 x 10-11 x 60 x 60
mass (60 kg) sitting
1.52
1.5 metres apart?
= 1.07 x 10-7 N
Value of r
• It is important to realise that the value
for r, the distance between two masses,
is the distance between the centre of
the two masses.
i.e. if we consider the force between
the Earth and the Moon then the value
of r is the distance from the centre of
the Earth to the centre of the Moon.
Example 2
• Taking the radius of the Earth to be 6.4 x 106 m, find
the force of attraction on a 250kg satellite that is
orbiting at a height of 36 000km above the Earth.
• (mass of Earth = 6.0  1024 kg)
•
1.
This question should be broken down into two parts.
First of all, find the distance, r, between the two
objects.
2. Use Newton’s Universal Law
Example 2
1. r
= radius of Earth + the height above the Earth
= 6.4 x 106 + 36 000 x 103
= 4.24 x 107 m
2. F
= Gm1m2
r2
= 6.67 x 10-11 x 6.0  1024 x 250
(4.24 x 107)2
=
55.7N
2012 Revised Higher
2013 Revised Higher
Open-ended question
Special Relativity
Time dilation
• Imagine a lamp which sends a
pulse of light at the same
time as producing a click.
• The light is reflected from a
mirror, at a known distance,
D, from the lamp.
• When it arrives back at the
lamp it produces a second
click.
• The total time will be: t = 2D
c
Time dilation
• Now imagine that the two lamps are moving at an
identical horizontal velocity.
• To an observer moving with the lamps nothing will
have changed.
• However, if there is a stationary observer watching
the lamps move he will see the pulses of light take a
different path and move a longer distance, 2h.
Time dilation
• The time between clicks in this case will be:
• t = 2h
c
• Therefore, time will be different for two
observers watching an identical system (as h
is clearly bigger than D).
What is meant by time dilation?
• The time observed in a moving system will always
be greater than that measured in the stationary
frame of reference.
• Time dilation is a difference in a time interval as
measured by a stationary observer and a moving
observer.
• ie a stationary observer will record a greater time
than a moving observer for the same journey
travelling at speeds close to the speed of light.
Equation for time dilation
• t’ =
t
.
√1 – v2
c2
• t’ = time reference for the stationary
observer
• t = time reference for the moving observer
• v = velocity of moving observer
• c = 3 x 108 ms-1
• NB: v is often given as a unit of c i.e. 0.7c. In
this case v = 0.7 and c = 1
Example 1
• A spacecraft leaves Earth and travels at a constant speed
of 0.6c to its destination. An astronaut on board records a
flight time of 5 days.
• Calculate the time taken for the journey as measured by
an observer on Earth.
• t’
=
t
.
√1 – v2
c2
• t’
=
• t’
= 6.25 days
5 .
√1 – 0.62
12
Example 2*
• A rocket leaves a planet and travels at a constant speed of
0.8c to a destination. An observer on the planet records a
time of 20h.
• Calculate the time taken for the journey as measured by
the astronaut on board.
•
t’
=
•
20
=
t
.
√1 – v2
c2
•
t .
√1 – 0.82
12
20 x √1 – 0.82
12
t = 20 x (0.6)
•
t
•
= 12 h
= t
2012 Revised Higher
C
2014 Revised Higher
B
The Doppler Effect
What is the Doppler effect?
• The Doppler effect is the change in frequency you
notice when a source of sound waves is moving
relative to you.
• When the source moves towards you, more waves
reach you per second and the frequency is increased.
• If the source moves away from you, fewer waves
reach you per second and the frequency is
decreased.
• Doppler Shift Demonstrator
Calculating the frequency
Moving towards the source
• The observed frequency, fo, is higher:
• fo = fs
•
•
•
•
v .
(v - vs)
fs = frequency of source
v = speed of sound (approx 340ms-1)
vs = speed of source
Towards = Take away
Calculating the frequency
Moving away from the source
• The observed frequency, fo, is lower:
• fo = f s
v .
(v + vs)
• Away = Add
Example 1
• What is the frequency heard by a person driving at 15 ms-1
toward a blowing factory whistle (f = 800 hz) if the speed
of sound in air is 340 ms-1?
•
fo = fs v .
(v - vs)
= 800
340
.
(340-15)
= 800 x 1.04
fo = 837 Hz
Example 2
• What frequency would he hear after passing the factory
if he continues at the same speed?
•
fo = fs v .
(v + vs)
= 800
340
.
(340+15)
= 800 x 0.931
fo = 745 Hz
2013 Revised Higher
C
Redshift
Background information
• White light (light from galaxies and
stars) is broken up into all the colours
of the rainbow
• Red Orange Yellow Green Blue Indigo Violet
• Longer λ
shorter λ
• All
the
colours
wavelengths
have
different
What is redshift?
• Redshift (also known as Doppler shift) is how much the frequency
of light from a far away object has moved toward the red end of
the spectrum.
• It is a measure of how much the ‘apparent’ wavelength of light
has been increased.
• It has the symbol Z and can be calculated using the following
equation:
• Z = λo – λr
λr
it can also expressed as: Z = λo
λr
• λo = the wavelength observed
• λr = the wavelength at rest
-1
What is a blueshift?
• When we use the equation for redshift,
we can sometimes end up with a –ve
value.
• This means the object is moving closer
to you and is said to be blueshifted.
• It is a measure of how much the
‘apparent’ wavelength of light has been
decreased.
Redshift and velocity
• We can also work out the redshift if we
know the velocity that the body is
moving at (for slow moving galaxies):
• Z=
v
c
Wavelengths
• With a redshift, moving away, the
wavelength increases.
• With a blueshift, moving towards, the
wavelength decreases.
Example 1
• Light from a distant galaxy is found to contain the spectral lines
of hydrogen. The light causing one of these lines has (an
observed) measured wavelength of 466 nm. When the same line
is observed (at rest) from a hydrogen source on Earth it has a
wavelength of 434 nm.
(a) Calculate the Doppler shift, z, for this galaxy.
(b) Calculate the speed at which the galaxy is moving relative
to the Earth.
(c) In which direction, towards or away from the Earth, is the
galaxy moving?
(a)
Z
Z = 0.074
= λo – λ r
λr
= 466 - 434
434
Example 1
(b)
Z=
0.074 =
v
c
v .
3 x 108
v = 2.21 x 107 ms-1
(c)
Z is positive therefore galaxy is moving
away
Example 2
•
(a)
(b)
A distant star is travelling directly away from the Earth at a
speed of 2·4 × 107 ms1.
Calculate the value of z for this star.
A hydrogen line in the spectrum of light from this star is
measured to be 443 nm. Calculate the wavelength of this line
when it observed from a hydrogen source on the Earth.
(a)
Z = v / c = 2.4 x 107 / 3 x 108 = 0.08
(b)
Z = λo - 1
λr
0.08 = (443x10-9) – 1
λr
0.08 + 1 = (443x10-9)
λr
λr
= (443x10-9)
0.08 + 1
λr
= 410 x 10-9 m / 410 nm
Revised Higher 2013
A
Revised Higher 2014 Qu: 25(b)
Revised Higher 2014 Qu: 25(b)
Specimen Paper Qu: 6(a)
Solution
Hubble’s law
Hubble’s Law
• The astronomer Edwin Hubble noticed in
the 1920s that the light from some
distant galaxies was shifted towards
the red end of the spectrum.
• The size of the shift was the same for
all elements coming from the galaxies.
• This shift was due to the galaxies
moving away from Earth at speed.
The bigger the shift the faster the
galaxy moves
• Hubble found that the further away a
galaxy was the faster it was travelling.
• The relationship between the distance
and speed of a galaxy is known as
Hubble’s Law:
v = Ho d
• Ho = Hubble’s constant = 2.3 x 10-18 s-1
Hubble’s Constant
• The value of
Ho = 2.3 x 10-18 s-1 is
given in data sheet (and
is the value you would
use in an exam) but
can vary as more
accurate measurements
are made.
• The gradient of the
line in a graph of speed
v distance of galaxies
provides a value for
Hubble’s constant.
Example 1
• What is the speed of a galaxy relative
to Earth that is at an approximate
distance of 4.10 × 1023 m from Earth?
• v = Ho d
• v = 2.3 x 10-18 x 4.10 x 1023
• v = 9.43 x 105 ms-1
What is a light year?
• Sometimes distances can be given in light
years.
• One light year is the distance travelled by
light in one year.
• It can be calculated as follows using d = vt:
• 3 x 108 (speed of light) x 365 (days) x 24
(hours) x 60 (mins) x 60 (s)
• One light year = 9.46 x 1015 m
Specimen Paper Qu: 6(b)
2012 Revised Higher
E
Revised Higher (specimen paper)
• v = Ho d
• 5.5 x 105 = 2.3 x 10-18 x d
• d = 5.5 x 105
2.3 x 10-18
• d = 2.39 x 1023 m
The expansion of the
universe
Is the universe expanding?
• The Universe has been expanding since
the Big Bang.
• The objects within the Universe (ie
galaxies) are not expanding – rather the
space between them is.
• The expansion of the Universe is
actually accelerating.
• All galaxies are moving away from each
other
What evidence is there to support this?
•
•
We can estimate how far away
something is by its brightness –
exploding stars in our galaxies are
moving further away at an accelerated
rate.
Furthermore, light from some distance
galaxies was observed to display a
redshift, showing they were moving
away at speed.
How can you estimate the mass of a galaxy?
•
•
•
•
Measurements of the velocities of galaxies
and their distance from us lead to the
theory of the expanding Universe.
Gravity is the force which slows down the
expansion.
The eventual fate of the Universe depends
on its mass.
The orbital speed of the Sun and other
stars gives a way of determining the mass
of our galaxy.
How is the mass of the universe related
to its expansion?
•
•
•
It was thought that the gravitational
pull of all the mass in the universe
would slow the expansion.
However as the universe expands, the
force of gravity is decreasing.
Additionally there must be another
invisible force pulling the universe –
this is know as ‘Dark Energy’.
What is Dark Energy?
•
•
Measurements of the mass of our galaxy and
others lead to the conclusion that there is
significant mass which cannot be detected
— dark matter.
Measurements of the expansion rate of the
universe lead to the conclusion that it is
increasing, suggesting that there is
something that overcomes the force of
gravity — dark energy
Open-ended question
Specimen paper
Possible answer
• Yes, the universe is getting even bigger. Astronomers believe
that the universe is expanding - that all points in the universe
are getting farther apart all the time just like a balloon blowing
up.
• By drawing galaxies on the balloon, you can observe that it is not
the galaxies themselves that are expanding (as you blow the
balloon up over time) but the space between them, just like in
the universe (over time).
• It can also be used to work out speed of each galaxy by
recording the distances between galaxies and the time taken to
increase these distances when blown. The more you blow up the
balloon – the faster the speed ie acceleration.
• However, this model shows how the expansion works but it is
only a 2D model whereas the universe is 3D.
Big Bang Theory
What happens to the colour of
objects as they are heated?
• When an object is heated it does not initially
glow, but radiates large amounts of energy as
infrared radiation. We can feel this if we
place our hand near, but not touching, a hot
object.
• As an object becomes hotter it starts to glow
a dull red, followed by bright red, then
orange, yellow and finally white (white hot).
At extremely high temperatures it becomes a
bright blue-white colour.
Light and temperature
• We can see that the temperature of an
object affects the light it gives off.
• This means that the temperature of an
object is linked to both the frequency
and wavelength of the light it emits.
• A graph of intensity versus wavelength
has a characteristic shape and can be
shown in a “Planck distribution.”
Planck distribution
•
Is also known as a black-body
spectrum and has three main
features:
1.
The basic shape is more or
less the same
As the temperature of the
object increases, the peak
intensity wavelength
decreases (so frequency
increases)
As the temperature of the
object increases, the intensity
and energy increases
2.
3.
•
•
Summarised as follows:
T ↑ λ ↓ therefore f ↑ intensity
↑ energy ↑
2014 Revised Higher
B
What is the Big Bang
Theory?
The Big Bang
What is the Big Bang Theory?
•
•
•
•
•
•
The Big Bang Theory took place around 13.8 billion
years ago.
The universe was originally very hot and very dense
concentrated in a tiny point known as a singularity
(smaller than an atom).
It caused our universe to expand suddenly from the
singularity bringing time and space into existence.
Following the Big Bang, temperatures rapidly cooled
and tiny particles of matter began to form.
The first atoms to form were hydrogen and helium.
This matter created stars, galaxies and planets.
What evidence is there for the
Big Bang?
•
There are a number of factors that
provide evidence for the Big Bang:
1. The expanding universe
•
•
•
Light from the stars in the galaxy is
enough to light our sky at night.
The only explanation that it is not is
that the stars are moving away from
us.
An expanding universe must have
started out infinitely small and agrees
with the Big Bang Theory
2. Cosmic Microwave Background
Radiation
•
•
•
•
This radiation can be detected on Earth
coming from all directions in space.
This radiation is constant throughout the
universe.
As the universe expands the wavelength of
the radiation emitted increases (less
frequency) down to the microwave region.
This
radiation
provides
a
constant
temperature throughout the universe
(approximately 3K).
3. Abundance of light elements
•
•
The universe has an abundance of light
elements such as Hydrogen and
Helium.
As the universe grew these elements
fused together to create the other
known elements.
2014 Revised Higher
D
2012 Revised Higher – Qu: 22 (1/2)
2012 Revised Higher – Qu: 22 (2/2)
Solution
Solution
2013 Revised Higher – Qu: 25
Solution (a)
Possible solution (b)
• Light travels at approximately 3 x 108 ms-1
• One light year is the distance travelled by light in one
year – 3 x 108 x 365 days x 24 hours x 60 mins x 60
seconds = 9.46 x 1015 m.
• As stars in our galaxy can be well over hundreds and
thousands of light years away the time it takes their
light to reach us is massive.
• For example our nearest star (apart from the Sun) is
over 4 light years away, this means that the light we
see from this star was sent over 4 years ago.
• Effectively, we are looking back in time.
Open-ended question
2014 Revised Higher
Possible answer
• Yes the Universe is expanding. We know this because of:
1. Stars / galaxies are moving away from us;
2. Cosmic microwave background radiation; and
3. The abundance of light elements
• It is expanding in all directions with the space between each
galaxy increasing as they move away from each other
• “as fast as it can go, at the speed of light” – this lyric is wrong
as it suggests a constant speed however the expansion is
actually accelerating.
• “12 million miles a minute” - using 1610 metres per mile this
converts to 1.932 x 1010 metres a minute which is 3.22 x 108 m/s
and this is slightly more than the accepted value for the speed
of light.
• “the fastest speed there is” – this is true as nothing can travel
faster than the speed of the light