5.4.1 Point and spherical masses

Firstly, it is important to understand that ALL objects with mass create a gravitational field. This field extends to infinity but gets weaker with distance from the centre of mass of the object. Any object placed in a gravitational field will experience an attractive force towards the centre of mass of the object that is creating the field (e.g for objects on Earth we call this gravitational attraction the objects weight).

Gravitational field strength
The gravitational field strength 'g' at a point within a gravitational field is defined as the gravitational force exerted per unit mass on a small object placed at that point within the field. It is a vector and always points to the centre of mass of the object creating the gravitational field. It can be calculated using the equation:

g = F/m

NOTE: F is the gravitational force and m is the mass of the object in the gravitational field. Gravitational field strength has the unit  N kg^-1. This unit is the same as m s^-1. This is because gravitational field strength at a point is the same as the acceleration of free fall of an object at that point (g = a).

Gravitational field patterns
We can map the gravitational field pattern around an object by using gravitational field line/lines of force. The lines never cross and always show the direction of the field (the direction of force on a mass at that point in a field). Field lines that are closer together indicate a stronger field. The field lines around a spherical mass form a radial field. The gravitational field strength decreases with distance from the centre of the mass (the field lines get further apart). We can model large planets as point masses with field lines converging at the centre of mass of the object because the radial fields for a spherical mass and a single point mass are very similar. If the field lines are parallel and equidistant the field is uniform. In a uniform field the gravitational field strength does not change.

Gravimetry
This isn't exactly in the spec but we've covered it so I thought it might be useful to cover on the blog:

Gravimetry is the precise measurement and study of a gravitational field. Earth's gravitational field can be mapped and minute variations detected (usually due to topography, e.g. mountains, craters from meteorites etc). Gravimetry is used in mineral prospecting as denser rock causes a higher than normal gravitational field on the Earths surface (indicating metal ores are present there). We also use this technique to search for oil as oil-bearing rocks have a lower density than the surrounding rock.

Superconducting gravimeters are the most accurate gravimeters. They use liquid helium to cool a superconducting sphere in a magnetic field. The weight of the field is balanced by the effects of the magnetic field. The electric current required to generate the magnetic field depends on the Earth's gravitational field strength at that point.

Different fields
It is important to remember that not just gravitational fields give rise to a force. We also have magnetic, and electric fields we cover in this spec!

5.5.3 Cosmology

So in this section we'll use different unit for distance (as apposed to the meter). This is because the distances are so vast. The new units are as follows:

• Astronomical unit (AU)
• This is the average distance from the Earth to the Sun. It is most often used to express the average distance between the Sun and other planets in our Solar System.
• It is equal to 1.50 x 10¹¹ m
• Light-year (ly)
• This is the distance travelled by light in a vacuum in a time of one year. It is often used when expressing distances to stars or other galaxies.
• It is equal to 3.00 x 10⁸ x (365 x 24 x 60 x 60) = 9.46 x 10¹⁵ m
• Parsec (pc)
• So for this one, we need to be aware that in cosmology we measure angles in arcseconds and arcminutes. One arcsecond is 1/3600th of a degree (1 arcminute is 1/60th of a degree).
• The parsec is defined as the distance at which a radius of one AU subtends an angle of one arcsecond.
• The value of 1pc can be determined by tan(1/3600) = 1AU/1pc. This means that 1pc = 1AU/tan(1/3600). This equals 3.1 x 10¹⁶ m.
• It is important to realise that at a distance of n parsecs, the angle subtended by a radius of 1AU = 1/n arcseconds.

There is a special technique used to determine the distance to stars that are relatively close to Earth (like, less than 100 pc). This is known as stellar parallax. Parallax is the shift in position of a relatively close star against the backdrop of much more distant stars as Earth orbits the Sun. If p (parallax angle) is measured in arcseconds, the distance to the nearby star in parsecs is given by the following equation:

d = 1/p

This technique can't be used to measure the distance between stars bigger than 100pc from the Earth because as d increases the parallax angle decreases. Eventually becoming too small to measure accurately, even with the most advanced astronomical techniques.

The Doppler effect
The Doppler effect is used to determine the speed of moving objects.

When a wave source moves relative to an observer, the frequency and wavelength of the waves received by the observer change compared with what would be observed without relative motion. Originally, two points equidistant from the source would receive waves at the same frequency and wavelength as they were emitted from the source. When the source moves closer to one point, the waves received by this point will be compressed. They have a shorter wavelength and a higher frequency (therefore, a shorter period).

How fast the wave source moves relative to the observer affects the size of the observed shift in wavelength and frequency. For electromagnetic waves we can use the Doppler equation. The equation shows that the faster a source moves, the greater the observed change in wavelength and frequency:

Δλ/λ ≈ Δf/f ≈ v/c

NOTE: The Doppler equation can only be used for galaxies with speed far less than the speed of light.

As we know already, one technique to analyse starlight involves looking at the absorption spectra from stars. The Doppler effect can be used to determine the relative velocity of a distant galaxy. Any difference in the observed wavelengths of the absorption lines must be caused by the relative motion between the galaxy and the Earth.

NOTE: if the galaxy is moving towards the Earth the absorption lines will be blue-shifted (they move toward the blue end of the spectrum because the wavelength appears shorter). If the galaxy is moving away from the Earth the absorption lines will be red-shifted as the wavelength appears stretched.

Using data from the absorption spectra of many distant galaxies, Hubble made two key observations:

1. A confirmation that earlier observations that the light from the vast majority of galaxies was red shifted (they had a relative velocities away from the Earth)
2. He found that in general the further away the galaxy was the greater the observed red shift and so the faster the galaxy was moving

From these observations Hubble formulated his law (Hubble's Law): The recessional speed (v) of a galaxy is almost directly proportional to its distance (d) from the Earth.

This means that a graph of recessional speed against distance for all galaxies will produce a straight line graph through the origin. The gradient is a constant of proportionality - the Hubble constant (Ho). It's (current) value is 2.2  × 10^-18 km s^-1 Mpc^-1. From Hubble's law it can be derived that:

v ≈ Ho x d

Hubble's law has been very useful in determining key evidence for the Big Bang theory and the model of the expanding Universe (following the Big Bang). This model is the accepted explanation of the observation that the light from nearly all the galaxies we can see is red-shifted. The further two points are apart the faster their relative motion (the more red-shifted their spectra).

The cosmological principle is the assumption that (when viewed on a large enough scale) the Universe is homogenous and isotropic and the laws of physics are universal:

• Homogenous means the matter is distributed uniformly across the Universe (ie the density of the universe is uniform
• Isotropic means that the Universe looks the same in all directions to every observer (ie there is no end to the Universe)
• The laws of physics can be applied across the Universe (meaning that theories/models tested on Earth can be applied to everything within the Universe).

The Big Bang
Hubble's law and the microwave background radiation are two key pieces of evidence for the Big Bang theory. Hubble's law shows that space is expanding as the galaxies are receding from each other because space itself is expanding in all dimensions.

The existence of microwave background radiation is the second piece of evidence for the Big Bang. Microwave background radiation can only be explained by the Big Bang and the expansion of space. It's existence can be explained in two ways:

• When the Universe was young and extremely hot space was saturated with high-energy gamma photons. The expansion meant that space itself was stretched over time. The expansion stretched the wavelength of these high-energy photons so we now observe this primordial electromagnetic radiation as microwaves.
• The Universe was extremely hot and dense when it was young. Expansions over billions of years reduced it's temperature to about 2.7 K. The Universe can be treated as a black-body radiator. At this temperature the peak wavelength would correspond to about 1mm (in the microwave region of the spectrum).

We can estimate the age of the Universe by assuming that it has expanded uniformly over time since the Big Bang. This actually isn't the case lol. Results from recent observations show that the expansion of the Universe is accelerating. Nonetheless, this assumption will give a crude indication the Universe's age.

So, Hubble's law shows galaxies are receding from each other. If a galaxy at a distance 'd' is moving away at a constant speed 'v' then a time (d/v) must have elapsed since it was next to our galaxy. This time is roughly the age of the Universe. The ratio d/v is equal to 1/Ho meaning that:

age of the Universe 't' ≈ 1/Ho

NOTE: This gives the age of the Universe to be 4.5 x 10¹⁷ s (≈14 billion years).

As I mentioned above, it is now known that the Universe appears to be expanding at an increasing rate. The most widely accepted theory includes the concept of dark energy. It is suggested that this hypothetical form of energy fills all of space and tends to accelerate the expansion of the Universe. The two most significant discoveries that have changed our understanding of the Universe is the discovery of dark energy and dark matter.

We need energy to accelerate things. The term 'dark energy' was coined to describe a hypothetical form of energy that permeates all space. It is estimated that dark energy makes up around 68% of our Universe.

In the 1970s astronomers studying the Doppler shift in light from galaxies found that the velocity of the stars in the galaxies did not behave as predicted. It was expected that their velocity would decrease as the distance from the centre of the galaxy increases. This effect is observed in other gravitational systems where most of the mass is in the centre (e.g the moons of Jupiter). The observations can be explained if the mass of the galaxy is not concentrated in the centre. We currently think that there must be another type of matter which we cannot see. This dark matter is spread throughout the galaxy, explaining the observations. The Universe must be made up of 27% of this matter (according to calculations).

All we know about dark matter is we know it cannot be seen directly with telescopes and it neither emits not absorbs light.

The rest of the universe is made up of a small percentage of ordinary matter.

5.3.3 Damping

An oscillation is amped when an external force that acts on the oscillator has the effect of reducing the amplitude of its oscillations. E.g a pendulum moving through air experiences air resistance which damps the oscillations until the pendulum comes to rest.

When amping forces are small the amplitude of the oscillator gradually decreases with time but the period (T) is almost unchanged. This is light damping.

For larger damping forces the amplitude decreases significantly and the period also increases slightly. This is heavy damping (e.g is a pendulum was stopped with water). 

Very heavy damping occurs if we were to damp the pendulum with for example treacle/oil. Eventually there would be no oscillatory motion. Instead the oscillator would slowly move towards its equilibrium position.

The x-t graph is as follows:

When a system is displaced and allowed to oscillate without any external forces its motion is referred to as free oscillation. The frequency of the free oscillation is known as the natural frequency of the oscillator. A forced oscillation is one in which a periodic driver force is applied to an oscillator. The object will vibrate at the frequency of the driving force (the driving frequency).

If the driving frequency is equal to the natural frequency of an oscillating object resonance will occur. This causes the amplitude of the oscillations to increase dramatically and, if not damped, the system may break. The greatest possible transfer of energy from the driver to the forced oscillator occurs at the resonant frequency. This is why the amplitude of the forced oscillator is maximum. Resonance examples include:

• Tacoma Narrows Bridge
• Types of tuning circuits (e.g in car radios to select the correct frequency radio wave signal)
• Many clocks keep time using the resonance of a pendulum
• Musical instruments have bodies that resonate to produce louder notes
• MRI enables diagnostic scans of inside our bodies without surgery/the use of harmful X-rays (MRI stands for magnetic resonance imaging)

NOTE: damping a forced oscillation has the effect of reducing the maximum amplitude at resonance. The degree of damping also has an effect on the frequency of the driver when maximum amplitude occurs.

Amplitude-driving frequency graphs:

• Light damping: the maximum amplitude occurs at the natural frequency (fo) of the forced oscillator.
• As the amount of damping increases:
• the amplitude of vibration at any frequency decreases
• the maximum amplitude occurs at a lower frequency that fo
• the peak on the graph becomes flatter and broader

Image sources: Kerboodle OCR A Physics textbook

5.3.2 Energy of a simple harmonic oscillator

Okay so this is nice and short. It's just to do with the energy transfers involved in SHM.

As we know, energy must be conserved. Provided there are no frictional forces (e.g damping) the total energy for an object moving in SHM will remain constant:

• At the amplitude the pendulum has zero kinetic energy (it is very briefly stationary). All its energy is in the form of potential energy (GPE in the case of a pendulum, EPE in the case of a horizontal string)
• As it moves through its equilibrium position it has maximum kinetic energy (as it has maximum velocity) and no potential energy.

This graph nicely illustrates how the total energy remains constant:

Credit: Kerboodle OCR A Physics A level textbook

We also need to be able to interpret graphs. In this case I will use an example of a spring on a horizontal glider:

• EPE is given by the equation EP = 0.5kx² (k is the force constant of the spring. This means that a graph of EP-x will be parabolic
• The EPE is always positive and varies from 0 to 0.5kA²
• When at amplitude the glider will be momentarily stationary (so it has no kinetic energy) meaning the total energy of the oscillator equates to 0.5kA²
• The kinetic energy of the glider is the difference between the total energy and the EPE:
• EK = 0.5kA²- 0.5kx²= 0.5k(A²-x²)

5.3.1 Simple harmonic oscillations

Examples of oscillating motion include a simple pendulum, end of a ruler hanging over the edge of a desk, a volume of water in a U-shaped tube. In each example the object always starts in an equilibrium position and a force must be applied to displace it (it now begins to oscillate). When an object is displaced it travels back toward the equilibrium position at increasing speed (it is accelerating). It begins to slow down once it has reached the equilibrium position and eventually reaches maximum displacement. The maximum displacement is known as amplitude (the same as in waves). Again, it speeds up whilst returning to equilibrium position and then slows down to it's maximum negative displacement (etc, etc). There are a few key words we should cover before we delve into this topic:

• Displacement - the distance from the equilibrium position
• Amplitude - maximum displacement/maximum distance from the equilibrium position
• Period - the time taken for to complete one full oscillation
• Frequency - the number of complete oscillations occurring per unit time

NOTE: like in waves, we can also get phase difference with oscillating objects.


Angular frequency is a term used to describe the motion of an oscillating object and is closely related to the angular velocity of an object in circular motion. We can find it using the following equations:

ω = 2π / T

ω = 2πf

(T = the period of the oscillator, f is its frequency).

Simple harmonic motion
SHM is a kind of oscillating motion for which the acceleration is directly proportional to the displacement and acts in the opposite direction. In other words:

a = -ω² x

NOTE: the minus sign indicates that the acceleration of the object acts in the direction opposite to the displacement.

A graph of acceleration against displacement for an object moving in SHM has a gradient equal to -ω² (-angular frequency²). the gradient is constant implying that the frequency and period of an oscillating object in SHM is constant. It is also important to note that objects moving in SHM are isochronous, that is, the period T is independent of the amplitude A. This is because as amplitude increases speed increases, so the period does not change.

We need to know a quick experiment to determine the period and frequency of objects moving with SHM:
Using a pendulum:

• Attach a mass to a string on a clamp stand (ie, make a pendulum)
• Time the time taken to complete 10 oscillations on a stopwatch - divide this by 10 to obtain the period (T)/time taken to complete one oscillation
• Repeat with different amplitudes to show a consistent T. This proves that the period is independent of the amplitude.

Using a spring:

• Place a fiducial marker at the equilibrium position (this provides a clear point from which to start/stop time measurements)
• Repeat as above (take time for 10 oscillations etc).

Displacement-time graphs
The graph of displacement against time for an oscillator moving in SHM creates a sinusoidal shape. If no energy is transferred to the surroundings (ie damping) the amplitude will remain constant.

Lets take a pendulum as an example. At zero displacement the pendulum is at/moving through equilibrium position and the maximum/minimum displacement it is at the top/bottom of it's swing, respectively. The gradient is equal to the velocity (v) of the oscillator. At maximum displacement (amplitude) the velocity is zero as the mass is effectively turning around (it is momentarily stopped/turning and going back in the opposite direction, not sure if that made sense sorry). Velocity is maximum as the pendulum moves through its equilibrium position.

It is important to note that the velocity-time graph is similar to the displacement-time graph, only it looks like it's beed shifted along a bit (by T/4). The acceleration-time graph looks as though it has been shifted along by T/4 from the velocity-time graph, so by T/2 from the displacement-time graph. This makes it look the same as the displacement-time graph, only inverted (hence a = -ω² x ∴so a ∝ -x).

Photo credit: Kerboodle OCR A physics textbook

So, what do all these graphs actually mean? Well, we know that a mathematical definition of SHM is a = -ω² x. We know that displacement of a SHM oscillator varies with time in a sinusoidal manner (meaning that the x-t graphs can be a sine or cosine graph). The common solutions to the equation a = -ω² x are:

x = Acosωt

x = Asinωt

Now what's the difference I hear you shout. You use either one or the other, and which one you use depends on where the object begins oscillating from. If an object begins oscillating from it's amplitude use x = Acosωt. If it begins oscillating from it's equilibrium position, use x = Asinωt.

So, in SHM velocity varies with time. When angular frequency (ω) is increased with no change in amplitude the oscillator will be travelling the same distance in a shorter time interval. So Vmax increases (meaning the x-t graph gradient will increase).

Now consider increasing the amplitude. We know SHM is isochronous so it will travel the same distance in the same time period, so again velocity will increase. The velocity of a SH oscillator at displacement x is given as follows:

v = ± ω √(A²-x²)

We are aware that velocity has a direction as it is a vector so a velocity at any particular displacement will have a positive or negative value (depending on the direction in which the oscillator is moving). At the equilibrium position velocity is a maximum and x is zero:

v = ω A

5.2.2 Centripetal force

It is important to realise that when an object moves in a circular path it's direction is actually constantly changing meaning it's velocity is constantly changing. A change in velocity means that the object must be accelerating (the greater the change in velocity the greater the acceleration).

We already know that any accelerating object must be under the influence of a net/resultant force.Any force that keeps a body moving with a uniform speed (remember, this is not uniform velocity!) along a circular path is called a centripetal force. A centripetal force is always perpendicular to the velocity of the object meaning that the force has no component in the direction of motion so no work is done on the object. As a result its speed remains constant.

Examples of centripetal force include gravitational attraction for a satellite in orbit, friction for a car going around a corner, and tension in the string of a yo-yo.

At any point on a circular path linear velocity is always constant to the tangent of the path. For an object moving in a circle at a constant speed we can calculate its speed using:

speed = distance/time

We know that in one complete rotation the distance travelled is the circumference of a circle (2πr) and the time is the period (T). This means that:

v = 2πr/T

We know that ω = 2π/T. It therefore follows that:

v = r ω

This means that for objects with the same angular velocity the linear velocity at any instant is proportional to the radius. Increasing the radius will increase linear velocity proportionally.

The acceleration of any object travelling in a circular path at a constant speed is called the centripetal acceleration. It also always acts towards the centre of the circle (like centripetal force). Centripetal acceleration depends on  the speed (v = r ω) of an object and the radius (r) of the circular path. The faster the object travels the larger the acceleration. The smaller the radius is the larger the acceleration. We can determine centripetal acceleration with the following equation:

a = v² / r

NOTE: we can combine this with v = r ω to give a = v = ω² r.

We can combine F = ma with a = v² / r to give an equation for centripetal force:

F = ma = (mv²) / r

This means that for a constant mass and radius the centripetal force (F) is directly proportional to v². Since v = r ω, we can also write the above equation as F = m ω² r. Remember this force is ALWAYS towards the centre of the circular path.

We need to know how to investigate circular motion. What we did in class was effectively to put a bung on the end of a piece of string and spin it around above our heads. Hold the string through a straw and suspend a mass from the other end. The weight will remain stationary is the force it provides (mg) is equal to the centripetal force making the bund travel in a circular path. If the centripetal force required is greater than the weight the mass will move upwards. With this set up we can investigate the centripetal force required for different masses/radii/speeds.

EXTRA:
This stuff isn't specifically in the specification but we covered it in class and it comes up pretty frequently in exam Qs so I thought i'd cover it on the blog. Just skip it if you cba to read lol. It's about sources of centripetal force.

There are many sources of centripetal force, such as tension in a string, gravitational attraction, and friction. We often get questions in exams where the surface is 'banked' (ie sloped) - such as a bike in a velodrome or a plane in the sky. Centripetal force can also be due to changes in the normal contact force when an object travels in a circular path. E.g when a ferris where rotates a net force is required to make you travel in a circular path (as apposed to no net force when your stationary as R = mg).

5.2.1 Kinematics of circular motion

Okay so for those of you who take maths you can skip this paragraph as i'm just explaining about the radian. The SI unit for an angle is the radian. A radian is the angle subtended by a circular arc with length equal to the radius of a circle. One radian is approximately 57.3°. There are 2π radians in a complete circle. We can determine the angle in radians using the following equation:

angle in radians = arc length/radius

To describe the motion of moving objects we need to be able to describe their linear motion and also their circular motion. Any object moving in a circular path moves through an angle θ in a certain time (t). This gives a method of describing movement in terms of angular motion - the object will have an average angular velocity. The angular velocity (ω) of an object moving in a circular path is defined as the rate of change of angle:

ω = θ/t

If the object completes one full circle the time (t) will equal one period (T). This means that the object will move through an angle of 2π radians:

ω = 2π / T

NOTE: angular velocity is measured in radians per second. We can also express ω = 2πf as frequency is 1/T.

There are a few different units we can use express angular velocity (such as ° s^-1, rev s^-1, rmp, rad s^-1). We should use radians per second.