Ian Wilkinson

HSC Physics Module 9.2 Summary

1. The Earth has a gravitational field that exerts a force on objects both on it and around it

Define weight as the force on an object due to a gravitational field

All masses have a gravitational field associated with them, which is an area in which other masses experience an attractive force towards the original mass. This is similar to the concept of an electric field around a point charge.Weight is the force on a mass due to the gravitational field of another mass. As it is a force, it is measured in Newtons (N). This is different to mass, which is a measure of the quantity of matter in a body, and is measured in kilograms (kg).

Analyse information using the expression to determine the weight force for a body on Earth and for the same body on other planets

According to Newton’s Second Law:

As g is the acceleration due to gravity of a mass, weight force can be determined by the following equation:

where: F = Force [N] m = Mass [kg] g = Acceleration due to gravity [ms-2]

Gather secondary information to predict the value of acceleration due to gravity on other planets

The acceleration due to gravity on other planets (g) can be derived by considering the formula for the gravitational force between two objects, which will be discussed in 9.2.3

By substituting

which leads to

Where: g = Acceleration due to gravity [ms-2] G = Universal gravitational constant = 6.67x10-11Nm2kg-2

M = Mass of the planet [kg] d = distance between the centres of masses

NOTE: d is the distance between the centres of masses, and thus is measured from the centre of planets, NOT from the surface. HSC questions will often give the height above the centre in the given data, but the radius of the planet must be added in order to use the above formula.

Explain that a change in gravitational potential energy is related to work done

Gravitational potential energy, or Ep, is a measure of the potential energy of a mass due to presence of a gravitational field. Recall that work done is the change in energy of a mass. Thus if there is a change in the Ep of a mass, work has been done on that mass. If the Ep of a mass is increased, the work has come from an external object applying force to the mass, such as lifting a pencil upwards off a

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table. When Ep is decreased, the work is done by the gravitational field upon the mass, and Ep is most commonly converted to kinetic energy.

Define gravitational potential energy as the work done to move an object from a very large distance away to a point in a gravitational field

The gravitational potential energy (Ep) can be calculated using the formula for work done:

by substituting

The following equation can be derived

There is a problem with this equation, as it is assumes that g is constant. This is not true however, as g is proportional to the inverse square of the distance between centres of masses.By substituting

And by recognising that potential energy is measured from centres of masses, not from the surface of Earth (i.e. h=d=r)

Which leads to

There is however a problem with the above formula. The formula for acceleration due to gravity was derived from Newton’s Law of Universal Gravitation. In that formula, force has a value of zero at an infinite distance from Earth. Thus it follows that the zero reference point for Ep must be at an infinite distance also, where a mass experiences no weight force. But as previously established, moving an object towards a mass leads to a decrease in Ep. If an infinite distance is taken as the zero point, moving an object towards a mass must result in Ep taking a value less than zero, that is, a negative value. Thus the formula for Ep is:

Where Ep = Gravitational potential energy G = Universal gravitational constant m1 = Mass of object 1 m2 = Mass of object 2 r = distance between centres of masses

As can be seen in this diagram, Ep increases towards zero as it moves away from a body. Due to the Law of Conservation of Energy, the kinetic energy and Ep add to zero, as per the diagram.

The formula can also be derived by considering that work done is the integral of the force equation in

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respect to distance. Therefore, by integrating Newton’s Law of Universal Gravitation, with limits from infinity to r:

Perform an investigation and gather information to determine a value for acceleration due to gravity using pendulum motion or computer-assisted technology and identify reason for possible variations from the value 9.8ms-2

Refer to practical 9.2.1d)METHOD

A retort stand with boss head and clamp was set up, and a string attached to a 200g mass was clamped tightly. The length of string was measured, held 10° to the vertical, and released. The time for ten oscillation periods was recorded, and then divided by 10. The method was repeated for various lengths of string from 0.2m to 1.0m.

DRAW DIAGRAM IF POSSIBLE For SAFETY, use small masses and small swings (i.e. small

angles) to minimise equipment damage and possible injury.RESULTS

The following formula was used:

From the results, T2 (period squared) was plotted against l (length), and the gradient was found. The value of g was calculated using the gradient and the above formula, which lead to the following equation.

The value obtained was 9.7m-2

ACCURACY The value obtained was close to the accepted value of

9.8ms-2. Possible reasons for variation include:o Inaccuracy of measuring the length of pendulumo Unreliability of using a stopwatch to measure period

length => susceptible to human erroro Minor factors that influence g e.g. altitude where

measuredRELIABILITY

The experiment was repeated by measuring the time for 10 periods, and an average was taken => also minimised human error of stopwatch

No outliers were present in data All points were close to the line of best fit => precision A gradient was measured => provides more reliable results

VALIDITY

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By measuring the gradient, systematic error (y-intercept) was eliminated

All other variables (e.g. location of experiment) were controlled

A small angle of oscillation was used to minimise the dampening of oscillation

The accuracy was limited by the reasons stated above The accuracy and validity could be improved by using data

loggers and sensors to measure period, and by using computer technologies to graph the result

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2. Many factors have to be taken into account to achieve a successful rocket launch, maintain safe orbit and return to Earth

Describe the trajectory of an object undergoing projectile motion within the Earth’s gravitational field in terms of horizontal and vertical components

Projectile motion is the motion of an object where gravity is the only force on the object. This includes anything thrown, dropped, or launched into the air and left to continue unpowered flight. So a rocket is not a projectile, as it is powered by thrust force.Recall that a vector can be resolved into its relevant dimensions. For projectile motion, the relevant dimensions are the horizontal and vertical component of an object’s velocity. After the projectile has been launched, there is no horizontal force (ignoring air resistance), and the only force in the vertical component is gravity. As a result, a projectile accelerates towards the ground at a rate equal to the acceleration due to gravity (g). The laws of motion can be applied to the horizontal and vertical components independently.Through mathematical analysis, the predicted trajectory of an ideal projectile is a parabolic shape.

Solve problems and analyse information to calculate the actual velocity of a projectile from its horizontal and vertical components using:

Remember to only substitute in values after an equation has been rearranged, and to also substitute the relevant units and dimensions for data values. Always check the units and dimensions as a check that your working is correct. Define the origin as the most mathematically convenient location.Projectile motion questions often involve finding the following data:

Velocity of an object at any point in time (most commonly initial and final) => use trigonometry to find horizontal and vertical components.

Range (horizontal distance travelled by a projectile) Maximum height (vertical velocity equals zero) Height at any time (vertical displacement) Time taken to reach ground

Describe Galileo’s analysis of projectile motion

Galileo postulated that all masses, regardless of their size, fall at the same rate (i.e. acceleration due to gravity is independent of mass).

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This is only true if air resistance is ignored, and so Galileo had difficulty demonstrating his postulate. He eventually tested and measured his postulate by not dropping objects, but by rolling balls of different masses down highly-polished inclines. The lower acceleration of the balls made it easier to measure, and he was able to use trigonometry to calculate the acceleration due to gravity on Earth. He also showed that the flight time of an object projected from the same height horizontally would be independent of its initial horizontal velocity.

Galileo’s analysis of projectiles also introduced the idea of inertial frames of reference and relativity. By considering cannonballs dropped from moving ships, he observed that whilst from the ship’s frame the ball drops vertically, to an observer the ball drops in a parabolic shape. He concluded that velocity can be resolved into horizontal and vertical components, and that a projectile moves due to acceleration in the vertical component and inertia in the horizontal component. He used this idea to prove the heliocentric model of the universe, as it explained why objects weren’t left behind as the Earth moved. This advance in scientific thinking challenged the Aristotelian idea of impetus, and changed the direction of scientific thinking, leading to Newton to develop his laws of motion and theories on gravitation.

Explain the concept of escape velocity in terms of the:

gravitational constant

mass and radius of the planet

Escape velocity is the initial velocity required by a projectile to rise vertically and just escape the gravitational field of a planet. An object at escape velocity would rise up, slow down, but would not return. Escape velocity can be calculated by considering the conservation of energy, and comparing initial energy to final energy. If a projectile is launched at escape velocity, its kinetic energy at an infinite distance would be zero.

Where:U = Total energy of projectile

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Uk = Kinetic energy of projectileUG = Gravitational potential energy of projectile.By conservation of momentum:

Therefore

Rearranging we get:

Where:v = escape velocity (ms-1)G = 6.67x10-11kg-1m3s-2 = Universal gravitational constantM = Mass of planetr = radius of planetAs can be seen, escape velocity depends on the gravitational constant, and the mass and radius of the planet. It is independent of any physical features of the projectile. If a projectile travels at a speed greater than escape velocity, it will also escape, and have some kinetic energy when it is at an infinite distance.Note that this equation should be derived in an exam.Also, note that escape velocity is a theoretical idea, and that it would be practically impossible to achieve from Earth’s surface. This is because Earth’s atmosphere would cause the projectile to heat up and potentially vaporise the projectile, and that the extreme g-forces created as the projectile is accelerated would crush anything inside the projectile. Escape velocity can be demonstrated by the slingshot effect, which is discussed later.

Outline Newton’s concept of escape velocity

Newton postulated that it is possible to launch a projectile fast enough so that it could escape Earth’s gravitational field. He considered a thought experiment of launching a projectile horizontally from a very tall mountain. If the projectile is launched at low horizontal velocities, it will fall to the ground in a parabolic path. As the initial horizontal velocity it increased, the projectile will reach further points on Earth’s surface. If it is launched at a fast enough horizontal velocity, the object will travel around the Earth, because as the object falls, the Earth’s surface curves away from the projectile. The object would follow a circular orbit.

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If the projectile is launched at faster velocities, it would follow an elliptical orbit around the Earth. If it is launched at a fast enough velocity, the projectile would follow a parabolic orbit and escape Earth’s gravitational field (hyperbolic orbit at higher velocities). This is Newton’s concept of escape velocity.

Therefore, Newton’s reasoning is that there is an initial velocity at which a projectile can be fired so that it will follow a bound orbit around Earth and not hit the ground. Furthermore, if a projectile exceeds a certain velocity (escape velocity), the projectile will follow an unbound orbit and escape Earth’s gravitational field.

Identify why the term ‘g forces’ is used to explain the forces acting on an astronaut during launch

The g-force is the ratio of a person’s apparent weight at a given time to their true weight. Apparent weight is the sensation of weight that a person feels, and is equal to the sum of contact forces resisting a person’s true weight. Another way of classifying g-forces is that it is a measure of acceleration forces, with Earth’s gravitational acceleration as the unit. A person at rest on Earth’s surface experiences a g-force of 1.The apparent weight experience by an astronaut is mg + ma, where as shown in the diagram below.

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Therefore, g-force in the vertical direction is given by:

It is much easier however to remember the definition of g-force as the ratio of apparent weight to true weight, as the above formula does not work for horizontal and circular acceleration.G-forces are used to explain the forces acting on an astronaut during launch as it provides a more accurate and relevant representation of the stress experienced by an astronaut. Astronauts have different masses, so the forces acting on each astronaut is different. But as g-forces are a ratio, g-force is independent of mass, and thus allows for easier comparison of forces.

Discuss the effect of the Earth’s orbital motion and its rotational motion on the launch of a rocket

Earth’s rotation around its axis provides an initial velocity for a rocket being launched into space, as the rocket has the same initial velocity, thereby providing additional velocity during take-off. Similarly, Earth’s orbital motion around the Sun provides more relative velocity. The Earth’s rotates at 1700kmh-1 around the equator, and orbits at 107 000kmh-1 around the Sun. Thus exploiting the orbital and rotational motion of the Earth allows rockets to be launched at a higher velocity relative to an external frame.To maximise the additional velocity from Earth’s rotational motion, launch sites are chosen as close to the equator as possible (e.g. Florida, French Guinea), and are launched east, as Earth rotates east. Launches are also taken at the right time of day and right time of year to maximise the initial velocity of the rocket.

These set periods are called launch windows, and vary depending on the mission. Launching a rocket during a launch window saves fuel

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(therefore less environmental harm), and maximises possible payloads for a rocket.

Analyse the changing acceleration of a rocket during launch in terms of the:

Law of Conservation of Momentum

forces experienced by astronauts

A rocket is propelled by the expulsion of combusted gas particles in the opposite direction to the rocket’s motion. By Newton’s Third Law, the force of the gas particles being expelled from the rocket is equal to the force of the gas particles on the rocket, and so the rocket accelerates in space.MomentumThe propulsion of a rocket can also be analysed by applying the conservation of momentum to the rocket system. Before a rocket launches, its momentum relative to Earth is zero, as it has no velocity. Therefore at any point in the flight, the net change in momentum for a given time period must equal zero by the Law of Conservation of Momentum.

The change in momentum of the rocket is equal and opposite to the change in momentum of the gas particles being expelled from the rocket. Note that the oxygen supply also has to be stored in the rocket, as there is no oxygen in outer space.

Forces experienced by astronautBy applying Newton’s Third Law to a rocket, we can also see that the force of the gas particles being expelled is equal and opposite to the force of the rocket.

Before launch, and astronaut experiences a force of 1g, as the rocket is resting on the ground. During launch, a rocket experiences both thrust and weight force, so the rocket accelerates according to Newton’s Second Law.

Where:T = Thrust force [N]m = mass of rocket system [kg]Before take-off, the mass of the fuel constitutes around 90% of the initial mass. As the fuel combusts and is expelled from the rocket, the mass of the rocket significantly decreases. Additionally, as a rocket’s altitude increases, the acceleration due to gravity decreases. If thrust force is constant, we can see from the above equation that the acceleration of the rocket increases until the rocket exhausts of fuel.

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Rockets that are launched into space are generally multi-stage rockets, i.e. the rocket has multiple fuel supplies and engines that operate in stages. Each stage is sequentially shutdown to avoid excessive peaks in g-force that could impact astronauts. After each stage shuts off, there is a brief moment of 0g (weightlessness), as no force is supplied. The next stage then initiates, and proceeds much like the previous stage.

G-forces above 4 are potentially unsafe, as it could lead to blackouts for positive g-forces (blood drains from head, causing unconsciousness), or red-outs for negative g-forces (blood rushes to head, causing excessive bleeding and brain damage). Astronauts also experience loss of peripheral and colour vision. If an astronaut is lying upwards perpendicular to the g-force on a body-contoured couch, up to 20 g may be tolerated, as blood flow to the head is less affected. The sequential shutdown of multiple engines also reduces the effect of an excessive change in g-force on an astronaut.

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Analyse the forces involved in uniform circular motion for a range of objects, including satellites orbiting the Earth

Uniform circular motion is the motion of an object travelling in a circle with constant linear orbital speed. The velocity of an object at any instant whilst in uniform circular motion is tangential to orbital path. As object’s velocity is changing direction whilst in circular motion, there must be an acceleration, and hence a force. The force is called centripetal force, and is directed perpendicularly to the object’s velocity, and directed towards the centre.

Note that centripetal force is not a true force, but the force required for an object to move in circular motion. In the above diagram, the required centripetal force is provided by the tension in the rope attached to the rock.The equation for centripetal force is

A satellite orbiting Earth in uniform circular motion also experiences a centripetal force, supplied by the gravitational force of the Earth on the satellite. Earth’s gravitational force acts perpendicular to the satellite’s velocity, causing the satellite to undergo centripetal acceleration, and move in circular motion. The gravitational force on a satellite moving in circular motion is constant however, and so for a satellite, we need to consider the required orbital velocity for uniform circular motion to occur. Orbital velocity can be calculated by equating centripetal force with gravitational force (see below).

Solve problems and analyse Remember that r is the distance between centres of masses. If the

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information to calculate the centripetal force acting on a satellite undergoing uniform circular motion about the Earth using:

data given has the height of the orbit above Earth’s surface, you need to add the radius of Earth for the equation to be correct.Also note that for a satellite, centripetal force is provided gravitational force, so centripetal force in this case can also be calculated by using Newton’s Law of Universal Gravitation.

Compare qualitatively low Earth and geo-stationary orbits

A low Earth orbit is an orbit higher than 250km, and lower than 1000km. Above 250km a satellite is much less affected by atmospheric drag. Below 1000km a satellite is below the Van Allen radiation belts, which pose risk to live space travellers and electronic equipment due to the high radiation of the belts.A geo-stationary orbit is at an altitude over the equator so that the period of the orbit exactly matches the period of Earth’s rotation. This allows the satellite to appear to be over a fixed point on Earth’s surface, and the receiving dish is points at a fixed point in the sky. The radius of a geostationary orbit is 42 168km (altitude 35 800km), which can be calculated using Kepler’s Third Law, and taking Earth’s rotation period as 23h, 56m, 4s (one sidereal day). The orbital velocity is approximately 11, 000kmh-1.Below is a comparison of the two orbits.

Low Earth orbit Geostationary orbit250km-1000km altitude 35 800km altitude above

equator~90mins period Period same as Earth’s rotationCheaper to launch More expensive to launchStronger signal with little delay Weaker signal with longer delaySubject to drag and orbital decay

Do not experience orbital decay

Closer, wider view of Earth’s surface

Limited view of Earth’s surface

Satellite can be placed in any orbit desired

Satellite can only be in one specific orbital path

Uses include geotopographic studies, studying weather patterns, military spying, and civilian surveillance

Communication satellites, weather monitoring, and information relay

Define the term orbital velocity and the quantitative and qualitative relationship between orbital velocity, the gravitational constant, mass of the central body, mass of the satellite and the radius of the orbit using Kepler’s Law of Periods

Orbital velocity is the instantaneous velocity of an object in circular motion along its path. This velocity needs to be maintained in order for the satellite to stay in its path. Orbital velocity can be calculate by considering velocity as displacement over time:

Where:v = orbital velocity [ms-1]r = radius of orbit from centre of central mass [m]T = period of orbit [s]Kepler’s Law of Periods is the following:

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Where:r = radius of orbit from centre of central mass [m]T = period of orbit [s]M = mass of central massRearranging the expression for orbital velocity to make T the subject, and then substituting into Kepler’s Law of Periods, we get the following:

Rearranging:

Where:v = orbital velocity [ms-1]G = 6.67x10-11kg-1m3s-2 = universal gravitational constantM = mass of central bodyr = distance between centre of masses [kg]The above expression can also be derived by equating centripetal force with gravitational force.Qualitatively, we can see that the square of the orbital velocity is directly proportional to the mass of the central body, and inversely proportional to the radius of the orbit. The constant of proportionality is the universal gravitational constant. Note that orbital velocity is independent of the mass of the satellite.

Solve problems and analyse information using:

Remember that r is the distance between centres of masses. If the data given has the height of the orbit above Earth’s surface, you need to add the radius of Earth for the equation to be correct.

Account for the orbital decay for satellites in low Earth orbit

Satellites in low Earth orbit have a maximum altitude of 1000km, and at this altitude are subject to interaction with the atmosphere, despite the extremely low density of the atmosphere at this altitude. As the satellite interacts with the atmosphere, the air particles exert a frictional force in the opposite direction to the satellite’s velocity, and a small lift force.

The atmospheric drag reduces the satellite’s velocity. Some of the

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kinetic energy of the satellite is lost to heat energy due to friction as the satellite slows down, and so the energy of the satellite reduces. By considering the equation for centripetal acceleration

we can see that as velocity decreases, so does the radius of the satellite’s orbit. As the satellite descends, its velocity actually increases, as it is accelerating towards Earth. Its total energy is less however, as its gravitational potential energy has decreased.In addition, the atmosphere is denser at lower altitudes. Thus the atmospheric drag on the satellite increases, and it decays at a faster rate. This continues until ~200km, where the heat generated by friction becomes great, and the satellite disintegrates.

Discuss issues associated with safe re-entry into the Earth’s atmosphere and landing on the Earth’s surface

When a spacecraft returns to Earth surface, the astronaut face many dangerous issues. These issues include heat, g-forces, ionisation blackout, and landing on Earth’s surface.HeatAs a spacecraft is travelling through Earth’s atmosphere, it has a velocity of around 30, 000kmh-1. The spacecraft has a high quantity of kinetic energy, and much of this energy is converted to heat energy during re-entry as a result of atmospheric friction. The heat can cause the spacecraft to reach extreme temperatures, which could cause the spacecraft to disintegrate.The heat can be tolerated by designing the spacecraft to have a blunt shape, as this dissipates some of the heat into the air. The space shuttle also presents its underbelly at 40° to Earth’s atmosphere for similar reasons. Spacecraft also have heat shields, such as ablative material, which sacrificially burns to carry away heat. On the space shuttle, insulating tiles made of glass fibre, though 90% of the composition is air. This gives them excellent thermal insulation properties, though they must be waterproofed between flights as the tiles are porous. More simply, the temperatures the space shuttle endures can be reduced by taking longer to re-enter, which lengthens the time that kinetic energy is converted to heat. This is achieved by the space shuttle through a series of sharp s-banking turns.G-forcesThe deceleration of a spacecraft during re-entry can cause g-forces of up to 20g, which is extremely dangerous for humans. The tolerance of humans to high g-forces can be reduced by having a transverse application of the g-forces (i.e. lying perpendicular to deceleration). This means the blood does not drain from the head. Additionally, the astronaut should be lying face-up, as an “eyeballs-in” application of g loads is easier to tolerate. Supporting the body also increases tolerance, so astronauts lie in contoured couches specifically designed for each astronaut.

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Ionisation blackoutThe extreme heat generated by a spacecraft during re-entry causes atoms in the air around the spacecraft to become ionised. Radio signals cannot penetrate this layer of ionised particles, and so radio communication is not possible. This can cause an issue if urgent radio contact is necessary during re-entry. This problem can be minimised by carefully planning the re-entry so that the astronauts are self-sufficient.Reaching the surfaceAfter passing through the atmosphere, the issue of landing on Earth’s surface still needs to be addressed. Early Russian cosmonauts would decelerate, then jump out of the spacecraft and land with a parachute. American astronauts deployed parachutes for the spacecraft, and landed in the ocean. The spacecraft lands on a runway by decelerating through a series of sharp S-turns with the nose at 40°, and then land on a runway.

Identify that there is an optimum angle for safe re-entry for a manned spacecraft into the Earth’s atmosphere and the consequences of failing to achieve this angle

A spacecraft starts re-entry by orbiting the Earth in a low Earth orbit. The astronauts will then retrofire of their rockets (i.e. position the rockets in front of the spacecraft), causing the spacecraft to lose energy, slow down, and descend. As the spacecraft enters Earth’s atmosphere, both lift and drag act on the spacecraft. If the spacecraft enters the atmosphere at an angle too steep, the g-forces and heat experienced by the spacecraft could destroy the spacecraft. If the angle is too shallow, the lift force will be too great, and the spacecraft may skip off the atmosphere instead of penetrating it, and not have enough fuel to re-attempt re-entry. Thus an optimum angle for safe re-entry of manned spacecraft exists, and is between 5.2°-7.2°.

Identify data sources, gather, analyse and present information on the contribution of one of the following to the development of space exploration: Tsiolkovsky, Oberth, Goddard, Esnault-Pelterie, O’Neill or von Braun

Wernher von Braun => some of his achievements are below. Developed high thrust engines using liquid fuel Responsible for the development of V2 guided missiles in

Germany Proved gyroscopes could help stabilise rockets After WWII, von Braun’s team continued research on the V2

rocket assembly, and test rocket for high altitude research He led the development of Redstone Rocket, which was used

for ‘first’ nuclear missile test He helped develop to put America’s first satellite, Explorer I,

into space in 1958 Led the development of the Saturn rocket, culminating in the

Saturn V rocket, which propelled Apollo-11 to the moon in 1969.

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Director of NASA’s Marshall space flight centre, which was instrumental in the space race.

He proposed the idea of manned space station and Mars mission

Developed the concept of orbital warfare and the Space Shuttle

Promoted public interest in space exploration, as the technical director of 38 Disney films

Sources of data include the NASA history page and New Scientist article.http://www.newscientist.com/blogs/culturelab/2009/11/hypocritical-or-apolitical-von-braun-deconstructed.htmlhttp://history.nasa.gov/sputnik/braun.html

Perform a first-hand investigation and analyse data to calculate initial and final velocity, maximum height reached, range and time of flight of a projectile for a range of situations by using simulations, data loggers and computer analysis

Refer to practical 9.2.2o)METHOD

The following website was used: http://zebu.uoregon.edu/nsf/cannon.html

The angle of initial velocity was set to 30°. The initial velocity of the projectile was varied, and the resulting maximum height, range, and time of flight were recorded in a table. This was repeated for angles of 45° and 60°.RESULTS

Refer to the results in the practical, though the results can easily be calculated if needed by using the kinematics equations

TRENDS: An increased initial velocity resulted in a greater range, time of flight, and maximum height reachedACCURACY/RELIABLITY/VALIDITY

The experiment was conducted as a computer simulation, so was very accurate/reliable/valid, as it modelled the physical application of the kinematics equations.

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3. The Solar System is held together by gravity

Describe a gravitational field in the region surrounding a massive object in terms of its effects on other masses in it

Recall that a field is a region in which a force is experienced, and that gravity is the force of attraction between masses. The gravitational field of a mass is the space surrounding it in which other masses experience a force of attraction due to gravitational force.Gravitational fields a vector fields; i.e. the field lines contain both a magnitude and a direction. The direction of the gravitational field is towards the centre of the respective masses, as gravity is a force of attraction. Whilst the field lines may look parallel close to the surface of a mass, they are in fact radial, and the spaces between field lines decrease the further away from a mass.

The gravitational field of an object theoretically extends to infinity, but as gravitational force is relatively very weak, it is only macroscopically observable for large masses for a finite distance. The field strength is the force per unit mass, and hence is equal to the acceleration of a mass due to gravity, or g.

Define Newton’s Law of Universal Gravitation:

The gravitational force of attraction between masses is proportional to the following values:

The mass in the centre of the field (directly) [m1 or M] The mass which is experiencing the force due to the

presence of the other mass (directly) [m2 or m] The inverse square of the distances between the two objects

[d]Thus Newton’s Law of Universal Gravitation is the following:

Where: F is measured in Newtons [N] m1 and m2 are measured in kilograms (kg) d is measured in metres [m] G is the gravitational constant (6.67x10-11 m3kg-1s-2)

As can be seen from the above expression, it is symmetrical about m1 and m2, which means they are interchangeable values. Hence Newton’s Law of Universal Gravitation is in fact an action-reaction pair which obeys Newton’s Third Law. The force of the first mass on

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a second mass is equal in magnitude to the force of the second mass on the first mass, but opposite in direction.Also note the relatively small value of G, which is why gravity is considered relatively a very weak force.

Present information and use available evidence to discuss the factors affecting the strength of the gravitational force

From the Law of Universal Gravitation

the strength of the gravitational force is dependent on the following. The mass of the two objects The distance between the masses.

In terms of Earth’s surface, the following can thus affect the strength of the gravitational force:

The altitude, as gravitational force decreases with an increased distance

Latitude => Earth is not a sphere, so gravity is strongest at the poles

Type of material below the surface, which alters the strength of gravitational force. For example, gravity is slightly stronger over land than over water.

Solve problems and analyse information using:

Remember to always make sure units and dimensions are balance. Also note that d is the distance between centres of mass, so radius of Earth must be added if the height above Earth’s surface is given.

Discuss the importance of Newton’s Law of Universal Gravitation in understanding and calculating the motion of satellites

Newton’s Law of Universal Gravitation has allowed us to have a more mathematical basis for understanding the motion of satellites. This is because gravity is the basis for the motion of satellites, in conjunction with other laws of motion. Hence the mathematical expression for gravitational force has allowed us to accurately calculate and predict the motion of satellites not only around Earth, but in the universe.Newton’s Law of Universal Gravitation has allowed us to calculate the orbital velocities required for specific satellite paths by the following derivation.First equate Newton’s Law of Universal Gravitation and centripetal force, which are equal for circular motion.

and Giving

The constant in Kepler’s Law of Periods can also be obtained by equating orbital velocity with

which is just the velocity of an object around the circumference of a

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circle. Newton’s Law of Universal Gravitation can also be used to derive the formula for gravitational potential energy, which was derived earlier.All of these expressions are very important in understanding and calculating the motion of satellites. Applications of this understanding include being able to send artificial satellites into specific orbital paths, and calculate the slingshot effect.

Identify that a slingshot effect can be provided by planets for space probes

The slingshot effect, also known as a gravity-assist manoeuvre or swing-by, occurs when a spacecraft is directed towards a planet at a high speed. The spacecraft enters the planet’s gravitational field, and then is ‘flung’ out at a higher speed, without expending much fuel.To further understand the slingshot effect, recall that orbits around object can be circular or elliptical. These are just two shapes an orbit can trace out; parabolic and hyperbolic orbits are also possible. As can be inferred, the orbits of masses trace out conic sections. At orbital velocity, the eccentricity of the orbit is zero, and hence a mass has a circular orbit. As its velocity increases, it traces out elliptical orbits with increasing eccentricity, until it reaches escape velocity, where the eccentricity is one, and the orbit is parabolic. Further increasing the velocity of the mass results in a path of greater eccentricity, and hence hyperbolic orbits occur. Circular and elliptical orbits are considered closed or stable orbits, whilst parabolic and hyperbolic orbits are unbound orbits. The kind of orbit in fact relies on the sign of the mechanical energy of the orbiting mass, but velocity provides a simplified explanation for the shapes of orbits.

During a slingshot manoeuvre, a spacecraft passes close to a planet, and enters a hyperbolic orbit due to the gravitational field. The spacecraft accelerates due to gravity as it enters the field, and decelerates as it departs the planet. This results in the spacecraft’s exit speed relative to the planet being theoretically equal to its entrance speed, but the direction of its velocity is different.From an external frame of reference however, such as the Sun, the speed of the spacecraft has changed. This is because the planet itself is moving at a velocity due to its orbit, and this relative velocity is added to the spacecraft’s velocity. If the spacecraft passes behind the planet, its velocity increases, and if it passes in front its velocity decreases.

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Thus the planet transfers energy to the spacecraft, and so the manoeuvre can be considered an elastic collision, as both momentum and kinetic energy are conserved. As the planet’s mass is so large however, the change in its energy is very small.The slingshot effect is normally achieved by the spacecraft approaching the planet at an angle to its orbit. The craft enter the orbit, swings around the planet, and exits at the same angle at which it entered. The final velocity of the spacecraft is given by the following equation, derived from the condition that the slingshot effect is an elastic collision.

The resultant gain in speed occurs with little expenditure of fuel, so is a useful manoeuvre for spacecraft travelling through space. The maximum final speed occurs when the spacecraft approaches and exits parallel to the planet’s orbit.

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4. Current and emerging understanding about time and space has been dependent upon earlier models of the transmission of light

Outline the features of the aether model for the transmission of light

Physicists in the 19th century observed light to act as a wave, and so believed that it propagated through a medium, which they named the aether. Scientists believed that the aether had the following properties:

It filled all space, as light travels everywhere It was transparent It had an extremely low density, hence was undetectable It was stationary It permeated all matter, and yet was permeable to material

objects It had great elasticity, otherwise energy would be lost during

the transmission of light over long distancesDescribe and evaluate the Michelson-Morley attempt to measure the relative velocity of the Earth through the aether

Theory behind the experimentPhysicists believed that the aether was stationary, and as such the Earth was moving through the aether at 30kms-1. If this was true, then a so-called aether ‘wind’ would be present on Earth. Under Galilean relativity, which dominated physics at the time, this wind would cause the speed of light to be different when measured from Earth’s frame of reference. This is because light would be either travelling with the wind, or against the wind, much like the difference between rowing a boat upstream or downstream. Many experiments tried and failed to detect the aether wind, but it was determined that the mechanisms weren’t sensitive enough.The Michelson-Morley ExperimentIn 1887, the following experiment was set up by Michelson and Morley in order to detect the aether wind:

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Michelson and Morley had set up an interferometer that would detect the interference pattern two light sources due to the aether wind. A beam of light from the light source was split at the half-silvered mirror (the beam splitter) in the centre of the apparatus. The first beam travelled through the mirror, and was travelling against the aether wind. The beam was then reflected by the mirror, and was travelling with the aether wind, until it reached the half-silvered mirror, and was reflected to the detector.The second beam was initially reflected perpendicular to the aether wind, and as such was travelling across the aether wind. This beam was then reflected, and travelled through the half-silvered mirror to the detector. Both of these beams had travelled the same distance in the apparatus’ frame of reference, but under the aether model, these two beams would have travelled at different velocities due to the aether wind, with the perpendicular beam travelling faster than the parallel beam. As such, it was expected that an interference pattern would be detected, as the beams of light would be out of phase with each other due to their different paths.The experiment was conducted on a pool of mercury, which dampened any external vibrations, and allowed the apparatus to be rotated. If the apparatus was rotated, different interference patterns should theoretically be detected under the aether model.Result and evaluationDespite Michelson and Morley conducted the experiment numerous times, no shift in the interference pattern that could be attributed to the aether wind was detected. They conducted the experiment at different times of the year and at different locations, but they still produced a null result.The experiment was significant because it changed the direction of scientific thinking on the model of light. Many scientists attempted to explain the null result by changing the aether model, such as introducing the aether ‘drag’. And whilst it did not disprove the existence of the aether, it led more scientists to be more willing to accept Einstein’s rejection of the aether model.

Gather and process information to interpret the results of the Michelson-Morley experiment

The result of the Michelson-Morley experiment was that no shift in interference pattern was detected within error when the apparatus was rotated through various angles, or the time of year it was taken. Whilst the null result did not entirely disprove the aether model of light, it did show that there was no evidence for the aether model of light, and led to scientists to reconsider whether the aether model was valid.I obtained information on the Michelson-Morley experiment from the Jacaranda Physics textbook, and from the University of Virginia website.

Discuss the role of the Michelson-Morley experiments in making determinations about competing theories

The consequence of the null result from the Michelson-Morley experiment was that it split scientific thinking on the nature of light waves. Some scientists still maintained the aether model, and believed that the model just needed improvement, as the aether model was consistent with the successful wave model at the time. Other scientists however believed that an alternative to the aether model was needed, as there was little evidence to support the

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predictions of the aether model.Einstein had been proposing an alternate model of electromagnetic radiation as a consequence of Maxwell’s equations and study into EMR. When Einstein proposed his special theory of relativity, he not only provided a competing theory to the aether model, he also challenged the Newtonian mechanics concept of an absolute frame of reference. Whilst scientists had initially been reluctant to accept Einstein’s model of electromagnetic radiation, the null result from the Michelson-Morley experiment aided scientists to become more willing to accept Einstein’s theory.

Outline the nature of inertial frames of reference

A frame of reference is anything with respect to which we describe motion and take measurements. Frames of reference can be classed into two groups: inertial and non-inertial frames of reference. An inertial frame of reference is one which is not accelerating, i.e. it is at rest or moving with constant velocity. Newton’s laws apply in an inertial frame of reference, but not in a non-inertial frame. In addition, Einstein’s special theory of relativity only applies to the special case of inertial reference frames; the general theory of relativity applies to non-inertial frames.Note that under special relativity, the Earth is a non-inertial reference frame due to the presence of gravity and Earth’s rotation. In calculations however, Earth is often approximated as an inertial frame.

Discuss the principle of relativity

The principle of relativity states that the laws of physics must be in the same form in all inertial frames of reference. This means that no inertial reference frame is truer than another, i.e. there is no absolute frame of reference. Hence if one event is observed from two inertial frames, both their observations are correct within their frame.It also means that there is no experiment that can be conducted inside an inertial frame to determine its state of motion (i.e. its velocity). The only way to determine velocity is to compare it to another frame of reference, and hence all motion is relative. For example, if you are sitting on a non-accelerating train, it is impossible to determine if you are moving or are stationary unless you look out the window, from which you can determine the relative motion of the train.One of the problems with the aether model was that it violated the principle of relativity. The speed of light within the frame would depend on its absolute velocity under the aether model, and thus a person would be able to determine the speed of the frame. This was one of the dilemmas for Einstein, and one aspect of his special theory of relativity.

Analyse and interpret some of Einstein’s thought experiments involving mirrors and trains and discuss the relationship between thought and reality

When Einstein was forming his theories on light and relativity, many of his postulates stemmed not from physical observations, but from thought experiments. One of these thought experiments involves a person travelling at the speed of light and looking at their reflection in a mirror.

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Einstein’s question was whether the person would be able to see their own reflection. Under the aether model, light would travel at the same speed as the train and would never reach the mirror; hence the person would see no reflection. But this would violate the principle of relativity, as then it would be possible to detect the motion of the train within the inertial frame.If the principle of relativity held, the person would be able to see their reflection, and hence the light in the frame would be travelling at the normal speed of light. But under Galilean relativity, an observer at rest outside the train would see light travelling at twice the speed of light.Einstein believed that this would not occur, and instead both people would see light travel at the speed of light (3x108ms-1). Thus he postulated that the speed of light is constant in all frames of reference.This postulate was based on logical thought, as the technological limitations of the time meant that Einstein had no way to physically measure his postulates. His thought experiments however allowed him to analyse and demonstrate his theory of relativity, even though they could not be tested in reality. From these thought experiments, Einstein was able to make deductions based on logic and fact to develop new theories, many of which have been proven physically correct since. Thus thought allows us to conduct hypothetical experiments using logic to gain a deeper scientific understanding, even if such experiments cannot be easily reproduced in reality.

Describe the significance of Einstein’s assumption of the constancy of the speed of light

One of the postulates of Einstein’s special theory of relativity is that the speed of light c (3x108 ms-1) is constant in all frames. But by considering the equation speed = distance/time, and applying it the thought experiment above, under Newtonian physics the observer would see light travel twice the distance in the same time, and hence have a velocity at twice the normal speed of light.For Einstein’s assumption of the constancy of the speed of light to be true, the length and time measured by each person must be different. Einstein thus deduced time and length are not absolute values, but relative depending on the frame of reference. This also supports the idea that there is no absolute frame of reference.

Identify that if c is constant then space and time become relative

Under Newtonian mechanics, space and time were absolute, and whilst motion was considered relative. But if we assume that the speed of light is constant, space and time must also be relative values too, leading to the concept of the space-time continuum. This means that all events in the universe must not only be defined by space, but also by time (i.e. in four dimensions). Such a definition

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allows an event to be fully defined in a frame of reference, and explains why both observers in Einstein’s thought experiment both measure the speed of light as 3x108ms-1.

Discuss the concept that length standards are defined in terms of time in contrast to the original metre standard

The metre was first defined in 1793 by the French government as the one ten-millionth (10-7) of the distance between the equator and the north pole, passing through Paris. Three platinum standards, and several iron copies were created, and this was considered the standard metre.This current definition of the metre was adopted in 1983 as the distance that light travels in 1/299 792 458 of a second (a second is defined as 9 129 642 770 oscillations of a Cs-133 atom). The need for this revision was because the initial definition of the metre was defined under the Newtonian assumption that space and time were absolute. But under special relativity, space and time are both relative, and so the current length standard takes the constancy of the speed of light into consideration.

Explain qualitatively and quantitatively the consequence of special relativity in relation to:

the relativity of simultaneity

the equivalence between mass and energy

length contraction time dilation mass dilation

The relativity of simultaneityThe relativity of time impacts on what we consider to be simultaneous events. For two events to be simultaneous, they must occur at the same time. But as time is relative, two events may be seen to occur at the same time in one frame, but at different times in a different relativistic frame, and hence meaning that simultaneity is also relative.One way to observe this concept is with another thought experiment. Consider a train travelling near the speed of light with a lamp in the centre of the carriage, and a front and back door.

The doors on the train are light-operated, so when the light from the lamp reaches the closed doors, they will open.In the frame of reference of the train, the distance to each door from the lamp is equal. Thus when the lamp is turned on, the doors will open at the same time, and will be a simultaneous event.A different situation will occur for a stationary observer outside however. Whilst the observer sees the light travelling at the same speed (c), the distance it travels is different due to the relativistic motion of the train. Hence the observer would see the two events of the back and front door opening as distinct in time, and would not

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be simultaneous. Both of these observations are correct, as space and time are relative, and there is no absolute frame of reference.The equivalence of mass and energyEinstein proposed in his special theory of relativity that energy and mass were equivalent, and that energy could be could be converted to mass and vice versa. Einstein reached this conclusion by considering the conservation of momentum on a relativistic scale, as time dilation would result in momentum not being conserved unless mass could be converted to energy. The energy-mass equivalence can also be seen in nuclear fusion reactions in stars, where the mass of the fuel (e.g. hydrogen) is converted to energy in the process of fusion. Energy and mass are equivalent by a factor of the speed of light (c) squared, leading to the equation:

Length contractionAs previously mentioned, space and time are relative quantities, and depend on the relative velocities of different objects. For space, length of an object contracts as measured from an external frame as its relative velocity increases. Quantitatively:

where:lv = the length of an object as measured from an external frame (will always be less than proper length)l0 = proper length = the length of an object as measured in its rest frame (i.e. the frame of the object)v = the relative velocity of the object to the external framec = 3x108ms-1 = speed of light in a vacuumNOTE: Length only contracts in the direction of relative motion. The length of an object perpendicular to motion does not contract.Time dilationIf the length of an object contracts at relativistic speeds, the time measured in an external frame must dilate (i.e. take longer) in an external frame in order to maintain the constancy of the speed of light. Hence as an object reaches higher speeds, the time measured from an external dilates. Quantitatively:

where:tv = time taken for an event to occur as measured from an external framet0 = proper time = time taken for an event to occur as measured from the rest frame of the objectv = the relative velocity of the object to the external framec = 3x108ms-1 = speed of light in a vacuum

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Mass dilationIn addition to length contraction and time dilation, the mass of an object dilates as it approaches relativistic speeds. Quantitatively:

where:mv = the mass of an object as measured from an external framem0 = rest mass = the mass of an object as measured in its rest framev = relative velocity of the object to the external framec = 3x108ms-1 = speed of light in a vacuum.

Consider in the above equation as v tends to c (i.e. the object approaches the speed of light). The bottom of the equation limits to zero, and so the relativistic mass approaches infinity. Thus it is impossible for an object with mass to reach the speed of light, as it would require infinite energy to do so. As can be seen, some of the energy that goes into accelerating an object to increase its kinetic energy is in fact converted to mass.

Solve problems and analyse information using:

Remember that length only contracts in the direction of motion. Also, remember that the length of an object contracts at relativistic speeds, whilst time and mass dilate. Take care to note what the rest frame of an object is and what the external frame is.

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Discuss the implications of mass increase, time dilation and length contraction for space travel

One significant implication of special relativity is to relativistic space travel. Whilst such space travel is not practically possible at the moment, its implications mean that such space travel would be subject to very different circumstances than have been currently experienced.Mass increaseAs a spacecraft accelerates to relativistic speeds, much of the energy supplied by the fuel converts to mass, increasing the mass of the spacecraft. As the mass of the spacecraft increases, more force is required to maintain a constant acceleration to reach relativistic speeds, and thus more work needs to be done on the spacecraft. Hence mass dilation means that it is also difficult to attain relativistic speeds, since more energy is required to accelerate the spacecraft at higher speeds. It would also be more costly to reach such speeds, as more fuel would be required.Another consequence of mass dilation is that the speed of light can never be achieved, as the mass of an object tends to infinity. In terms of the size of the universe, the speed of light is relatively small, and thus reaching even the closest stars would take at a minimum several years from Earth’s frame.For example, the closest star to Earth is Proxima Centauri, which is 4.3 light years away. As an object cannot travel faster than the speed of light, it would take at least 4.3 years in Earth’s frame to reach the star, and then at least 4.3 years to return. To reach the centre of the Milky Way, it would take 265 million years.Time dilationTime dilation causes time to be slower on board a spacecraft travelling at relativistic speeds. Time dilation means that it would be possible to achieve interstellar space travel within an astronaut’s lifetime, despite the longer time taken in Earth’s frame.For example, if a spacecraft where to travel to Alpha Centauri at relativistic speeds, it would take less than 4.3 years in the astronaut’s frame of reference.Length contractionLength contraction has the same implication on space travel as time dilation. From the spacecraft’s frame, the distance they are covering to the star is less as space is moving relative to the spacecraft, whilst time appears to be normal. Thus the spacecraft appears to travel less distance, and hence why it takes less time from the spacecraft’s frame. This again shows that space and time are relative, as time appears as normal on the spacecraft, even though it is slower in Earth’s frame.

Analyse information to discuss the relationship between theory and the evidence supporting it, using Einstein’s predictions based on relativity that were made

A successful scientific theory should be a successful explanation for a scientific idea or concept that can be demonstrated in practice, and be used to predict observations. A scientific theory should also be consistent with other accepted scientific theories and laws. Whilst most scientific theories are based on practical observations, it is not necessary for a successful theory to have practical evidence if it is

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many years before evidence was available to support it

not possible to obtain it. As long as the theory is a logical explanation for a scientific concept, and is consistent with other accepted theories, it can be considered a valid scientific theory, which can be confirmed later by evidence.Einstein’s special theory of relativity is one example a theory made without physical evidence. When Einstein proposed his theory in 1905, he did so without practical evidence, as there was no technology at the time able to prove his theories. It was not until several decades later that some of his theories were able to be proven, and even modern technology is incapable of testing many aspects of relativity. His theory has been successful however, as it provided a logical explanation to relativity, and was consistent with other theories at the time.There is however scientific evidence of special relativity that was discovered in the latter half of the 20th century, many years after Einstein first proposed his theory. The evidence includes atomic clock analysis and atmospheric mesons.Atomic clocksAtomic clocks that are accurate enough to measure time dilation at relatively small speeds have only been developed in the past few decades. The experiment involved calibrating two atomic clocks to the same time, and then placing one onto a jet plane. The plane flew around the world at high speed, whilst the other was stationary on Earth. When the other atomic clock returned, it was found that the one on the plane had run slightly slower, and hence time had dilated due to its relative velocity.MesonsAnother piece of evidence for time dilation is atmospheric mesons, which are particles produced in the atmosphere by incoming cosmic rays. In the laboratory, mesons only have a lifetime of 2.2μs, but when they are travelling to Earth at 0.996c, it would take them 16μs to reach Earth’s surface, where they have been detected. Thus in their rest frame, mesons have a lifetime of 2.2μs, but their relativistic speed causes time to dilate, and so is measured as 16μs, and is thus evidence for time dilation.As parts of Einstein’s theory of relativity have been proven many years after they were first proposed, we can see that practical evidence is not necessary for a successful scientific theory. Many modern scientific theories do not have evidence in reality, but are still accepted

Perform an investigation to help distinguish between non-inertial and inertial frames of reference

Refer to practical 3.2.4 l)METHOD

An accelerometer was attached to the back of a dynamics trolley, and the position of the liquid in the accelerometer was measured at rest. The accelerometer was then pulled at various accelerations and at constant velocity, and the resulting shape of the accelerometer was recorded.

The following set up can also be used

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DRAW DIAGRAM IF POSSIBLERESULTS

The accelerometer was flat at rest and at constant velocity The liquid was sloped downwards, and more liquid was

towards the back of the accelerometer as it accelerated in the positive direction

The liquid was sloped upwards, and more liquid was towards the front of the accelerometer as it accelerated in the negative direction

Refer to the practical report for the diagrams of the various shapes of the accelerometer.RELIABILITY/ACCURACY/VALIDITY

The experiment was repeated several times, and a general image was obtained

The experiment was able to distinguish between inertial and non-inertial frames, and hence tested the aim

The results corroborated with expected results The accuracy/validity of the experiment could have been

improved by using sensors and data loggers to detect the shape of the accelerometer.