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. m 1 and m 2 are the masses of the two objects, and r is the distance between them. G= gravitational constant= 6.67 x Nm 2 kg -2

Example 1 What is the force of attraction between two pupils of average mass (60 kg) sitting 1.5 metres apart? F = Gm 1 m 2 r 2 = 6.67 x x 60 x = 1.07 x 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 10 6 m, find the force of attraction on a 250kg satellite that is orbiting at a height of km above the Earth. (mass of Earth = 6.0  kg) This question should be broken down into two parts. 1.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 x 10 3 = 4.24 x 10 7 m 2.F = Gm 1 m 2 r 2 = 6.67 x x 6.0  x 250 (4.24 x 10 7 ) 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 – v 2 c 2 t’ = time reference for the stationary observer t = time reference for the moving observer v = velocity of moving observer c = 3 x 10 8 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 – v 2 c 2 t’ =5. √1 – t’ = 6.25 days

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’= t. √1 – v 2 c 2 20 =t. √1 – x √1 – = t 1 2 t = 20 x (0.6) t = 12 h

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, f o, is higher: f o = f s v. (v - v s ) f s = frequency of source v = speed of sound (approx 340ms -1 ) v s = speed of source Towards = Take away

Calculating the frequency Moving away from the source The observed frequency, f o, is lower: f o = f s v. (v + v s ) 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 ? f o = f s v. (v - v s ) = (340-15) = 800 x 1.04 f o = 837 Hz

Example 2 What frequency would he hear after passing the factory if he continues at the same speed? f o = f s v. (v + v s ) = (340+15) = 800 x f o = 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 R ed O range Y ellow G reen B lue I ndigo V iolet Longer λ shorter λ All the colours have different wavelengths

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 = λ o – λ r it can also expressed as: Z = λ o - 1 λ r λ r λ o = the wavelength observed λ r = the wavelength at rest

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 = λ o – λ r λ r = Z = 0.074

Example 1 (b) Z = v c = v. 3 x 10 8 v = 2.21 x 10 7 ms -1 (c)Z is positive therefore galaxy is moving away

Example 2 A distant star is travelling directly away from the Earth at a speed of 2·4 × 10 7 ms  1. (a)Calculate the value of z for this star. (b)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 10 7 / 3 x 10 8 = 0.08 (b)Z = λ o - 1 λ r 0.08 = (443x10 -9 ) – 1 λr λr = (443x10 -9 ) λ r λ r = (443x10 -9 ) λ r = 410 x m / 410 nm

Revised Higher 2013 A

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 = H o d H o = Hubble’s constant = 2.3 x s -1

Hubble’s Constant The value of H o = 2.3 x 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 × m from Earth? v = H o d v = 2.3 x x 4.10 x v = 9.43 x 10 5 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 10 8 (speed of light) x 365 (days) x 24 (hours) x 60 (mins) x 60 (s) One light year = 9.46 x m

Specimen Paper Qu: 6(b)

2012 Revised Higher E

Revised Higher (specimen paper) v = H o d 5.5 x 10 5 = 2.3 x x d d = 5.5 x x d = 2.39 x 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. yellowwhiteAs 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 2.As the temperature of the object increases, the peak intensity wavelength decreases (so frequency increases) 3.As the temperature of the object increases, the intensity and energy increases 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

2013 Revised Higher – Qu: 25

Solution (a)

Possible solution (b) Light travels at approximately 3 x 10 8 ms -1 One light year is the distance travelled by light in one year – 3 x 10 8 x 365 days x 24 hours x 60 mins x 60 seconds = 9.46 x 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 x metres a minute which is 3.22 x 10 8 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