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9.2 S p a c e. 1. Gravitational fields. Weight. Weight is the force on an object due to a gravitational field. The rest mass of an object remains constant irrespective of the gravitational field. The weight of an object changes as the gravitational field changes: F W = mg

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9 2 s p a c e
9.2 Space



Weight
Weight

  • Weight is the force on an object due to a gravitational field.

  • The rest mass of an object remains constant irrespective of the gravitational field.

  • The weight of an object changes as the gravitational field changes:

    FW = mg

    Where,FW = weight (N)‏

    m= mass (kg)‏

    g= gravitational acceleration (ms-2)‏

  • The mass of an astronaut at rest on Earth remains the same on another planet

  • The weight of an astronaut on another planet will depend on the planet's gravitational field


Calculating weight
Calculating weight

  • The reading on the scale is 57 kg. Although the scale actually measures weight (it measures a force), scales are traditionally calibrated in the units for mass.

  • The weight of the person on the scales can be calculated using :

    FW = mg

    FW = weight N; m = 57kg and g= 9.8ms-2

    FW = 57 x 9.8 = 558.6 N

  • The value of g on Venus is 8.9ms-2 . On Venus, this person would still have a mass of 57 kg but her weight would be 447 N (57 x 8.9 )‏


Gravitational potential energy
Gravitational Potential Energy

  • The change in gravitational potential energy is related to the work done:

    Work: W = Fs (force used to move an object from the surface to a height of s metres)‏

    In a gravitational field: F= Fw = mg

    Therefore W = mgs

    Potential energy near the Earth's surface is given by:

    Ep = mgh

    The two expressions are clearly identical

    W = Ep


Gravitational potential energy1
Gravitational Potential Energy

  • Gravitational potential energy is the work done to move an object from a very large distance away to a specific point in a gravitational field.

  • When a rocket is launched into space, work is done on the rocket hence the rocket's gravitational energy increases

  • When a meteor approaches the Earth its gravitational potential energy decreases (as the meteor speeds-up, potential energy is converted into kinetic energy )

  • The lowest Ep in the gravitational field surrounding a planet is at the surface of the planet.

  • The highest Ep is when that of object is at an 'infinite' distance from Earth. Its gravitational energy due to Earth's gravitational at infinity is zero.

  • It follows that all other values of Ep must be negative

G = 6.67300 × 10-11 m3 kg-1 s-2 (Universal gravitation constant)‏


E p example problem
Ep example problem

  • Calculate the energy that was required to move The Hubble Telescope (11 110 kg) to its parking orbit (559 km)‏

  • To do this we have to find the change in energy from its value on the surface to its value in orbit


Determination of g
Determination of g

To use a simple pendulum to determine the value of g.

  • The equipment used is shown to the left.

  • Displace the pendulum through a small angle ( < 10 o) and set it swinging.

  • Using a stop watch measure the time taken for ten periods (a period is a full swing from A to B and back to A)‏

  • Repeat a number of times and find the average period (T) for that length (L)‏

  • Repeat steps 3 to 6 for at least three different lengths.

  • Plot T2 vs. L.

  • The gradient of the line is: T2 / L

  • Using the formula adjacent and the gradient of the line, work out a value for g


Variations from 9 8 ms 2
Variations from 9.8 ms-2

  • The pendulum experiment generally yields fairly accurate results , ranging from 9.5 to 10 ms-2 for g.

  • Experimental errors occur in any experiment. In this case there are errors in measurement as well as errors inherent in the equipment used.

  • The average gravitational acceleration at the surface of the Earth is 9.80 ms-2. It can vary by a total of up to about 1%. This is largely due to a combination of 3 factors:

    Latitude:

    • Gravity is an average 9.78 ms-2 at the equator. It is about 0.5% higher at the poles at an average 9.83 ms-2. This is due to a combination of effects:

    • The spin of the Earth creates an outward force that is greatest at the equator. The difference is small, but enough to make the launch of space rockets cheaper near the equator than near the poles for most orbits.

    • The Earth's diameter at the equator is about 12,756 km . The distance from pole to pole is about 12,713 km This means that the surface at the equator is further away from the centre of the Earth.

      Elevation:

    • Gravity decreases with distance from the centre of the Earth. Gravity is about 0.2% lower at the top of Mount Everest than at sea level

      Crust thickness and Density:

    • Variations in the density and thickness of rocks in the Earth's lithosphere cause local variations in gravity. The denser the rock, the higher the value of g.


G for other planets
gfor other planets

  • The gravitational force exerted by a planet at its surface is related to its mass and its radius.

  • The radius and mass of a planet can be derived from astronomical observation.

  • g for any planet (including Earth) can be estimated using the formula:


Revision questions 1
Revision questions - 1

  • Define weight

  • Explain the relationship between a change in gravitational potential energy and the work done

  • Define gravitational potential energy

  • Describe an investigation you performed to determine a value for the acceleration due to gravity using pendulum motion

  • Identify reasons for possible variations of g from the value of 9.8 ms-2

  • Identify the data you would need in order to predict the value of g on other planets and describe how you would use it.

  • Calculate the work required to lift a 35 000 kg rocket to an altitude of 1 200 km

  • An astronaut lands on a new planet where the value of g is 10.67 ms-2. If the weight of the astronaut on Earth was 796 N determine his weight on the new planet



Projectile motion
Projectile motion

  • Projectile motion is considered as a motion in two dimensions: horizontal and vertical.

  • The horizontal velocity is constant if we ignore air resistance.

  • The vertical velocity is changed by the action of the acceleration due to gravity “g” which points directly downward at all times (toward the centre of the Earth or planet)‏

  • The trajectory is parabolic (if we ignore air resistance).‏

  • If the projectile starts and finishes at the same level, then the time to maximum height is at ½ of the time of flight.

  • At maximum height the vertical velocity is zero

  • If the initial velocity is upward and is taken as positive then downward is the negative direction. Hence g must be negative.


Projectile motion1
Projectile motion

  • The two velocities are the component vectors of the projectile velocity at any one time during the flight.

  • The horizontal component (x direction) remains constant throughout.

  • The vertical motion (y direction) is uniformly accelerated downward by -g.

  • At any time during the flight the velocity and/or the displacement of the projectile can be calculated by finding the vertical and horizontal components and then using Pythagoras and trigonometry

  • The time of flight is the same for both motions and is used to tie them together.


Projectile motion2
Projectile motion

  • Horizontal motion is described by:

    Where, t = time of flight; x = range

  • Vertical motion is described by:

  • Where at maximum height; uy = 0


Projectile motion a worked example
Projectile motion - a worked example

An object is fired at 100 ms-1 at an angle of 47o near the edge of a cliff face, which is 58 m above sea level. Find how far from the bottom of the cliff the object will enter the sea.

Step 1 do a sketch of the problem.

Step 2 resolve the initial velocity into the two components ux and uy

ux=100cos47 = 68.1998 ms-1 and uy=100sin47 = 73.1353 ms-1

Step 3 find the time to point H

vy=uy+ayt

0=73.1353 + (-9.8)t

t = 7.4628 s

Step4 find height of point H

vy2 = uy2 +2ayy

0 = (73.1353)2 + 2(-9.8)y

y = 272.8965 m

H from sea level = 272.8965 + 58 = 330.8965

Step 5 find time to impact from point H (y = - 330.8965 m)‏

y=ut+1/2at2

- 330.8965= 0t + ½ (-9.8)t2

t = 8.2176 s

Step 6 find the total time of flight

t = 7.4628 + 8.2176 = 15.6804 s

NB: the above can be done in one step using the 'quadratic formula' ony=uyt+1/2at2 with y = -58 and uy= 73.1353

Step 7 find the horizontal distance traveled

x=uxt x = 68.1988 x 15.6804

x =1069.38446352 =1070 m(do not round-off to significant figures until the final answer)‏


Projectiles an investigation
Projectiles – an investigation

  • An example of an investigation is the construction of a small catapult. You will need to research the subject and then come-up with your own design.

  • You will need to calibrate it by firing an object a number of times. Keep a full record of all your experiments in a log book.

  • On the side of your catapult you will need to attach some sort of scale that will allow you to bring back the arm reproducibly at different points to achieve different ranges.

  • Find the maximum range for your machine. Then try to set your arm so the it will propel the object ¾ of the way, ½ way and ¼ of the way. For each position validate your results by repetition.

  • For each position estimate angle of flight, maximum height time of flight using a camcorder or the video recorder on a mobile phone. From these calculate the initial velocity of the object

  • Next, set your self a fixed distance to aim at (e.g. 60% of your maximum range). Set your catapult using your calibrations and check how accurate you were.

  • Finally, write a report on your investigation. When you submit it for marking, include your log book.


Galileo galilei
Galileo Galilei

  • Before Galileo, it was believed that a projectile would travel in a straight line until it ran out of “impetus” and then it would fall straight down.

  • Using both experiment and mathematics, Galileo showed that the path of a projectile is a parabola.

  • Galileo rolled an inked bronze ball down an inclined plane onto a table. The ball thus accelerated, rolled over the table-top with uniform motion and then fell off the edge of the table. Where it hit the floor, it left a small mark. The mark allowed Galileo to measure the height and horizontal distance for different velocities.


Galileo galilei1
Galileo Galilei

  • Using a thought experiment, Galileo also showed that all objects will fall under the influence of gravity at the same rate irrespective of their masses (if there is no air resistance)‏

  • He didn't actually drop objects from the leaning tower of Pisa as legend describes.

  • Galileo's discovery was demonstrated by the Apollo mission astronauts in the vacuum of the Moon.

  • They dropped a feather and a hammer at the same time - as expected, both the feather and the hammer hit the lunar surface exactly at the same time! (A short clip of this is available on YouTube)‏


Escape velocity speed
Escape velocity (speed)

  • For a given gravitational potential energy, the escape velocity is the minimum speed an object (without propulsion) needs to have so that its kinetic energy equals the gravitational potential energy.

  • An object on the surface of a planet will need a kinetic energy equal to the gravitational potential energy at the surface of the planet. For the Earth's surface that kinetic energy results from a speed of about 11,100 ms-1

  • The concept does not directly apply to rockets as they are propelled by their engines and accelerating. But at the instant the propulsion stops, the vehicle can only escape if its speed is greater than or equal to the escape velocity for that position.

  • It would apply to an object fired by gun in an attempt to, say, reach the Moon. Much like Jules Verne's “moon gun” described in his 1865 novel From the Earth to the Moon.

  • Escape velocity is proportional to the square root of the planet's mass and inversely proportional to the square root of the distance from its centre, but is not affected by the object's mass.


Newton s escape velocity
Newton's escape velocity

  • Newton's thought experiment related to firing a gun pointing horizontally, on top of a very high mountain. At lower muzzle velocities the cannon ball would fall to the ground in a parabolic trajectory as shown by Galileo.

  • As the gun's muzzle velocity is increased the ball will hit further and further, but eventually, because the Earth is round it will end up not being able to hit the ground at all (because, as it falls, the surface of the Earth curves away from it.

  • The ball would complete a circular orbit around the Earth and if the gun was moved in time and there was no air resistance the ball would be in a circular orbit around the Earth

  • If the ball was fired at even higher speeds the orbit would become an ellipse.

  • At speeds above escape velocity the trajectory would become a hyperbola and the ball would leave forever.


G forces
G-forces

  • An acceleration of 1g is equal to standard gravity (9.80 ms-2)‏

  • G force = (a – g) /g

  • Human tolerances depend on the magnitude of the G-force, the length of time it is applied, the direction it acts, the location of application, and the posture of the body

  • A constant 16 g for a minute can be Fatal.

  • Astronauts and test pilots are tested for their ability to withstand high G's in special equipment.


G forces and the apollo space craft
G forces and the Apollo space craft*

1. Launch with ignition of the S-IC. Note how the acceleration rapidly rises with increasing engine efficiency and reduced fuel load.

2. Cut-off of the centre engine of the S-IC.

3. Outboard engine cut-off of the S-IC at a peak of 4g.

4. S-II stage ignition. Note the reduced angle of the graph for although the mass of the first stage has been discarded, the thrust of the S-II stage is nearly one tenth of the final S-IC thrust.

5. Cut-off of the centre engine of the S-II.

6. Change in mixture ratio caused by the operation of the PU valve. The richer mixture reduces the thrust slightly.

7. Outboard engine cut-off of the S-II at a peak of approximately 2.7g.

8. S-IVB stage ignition. Note again the reduced angle of the graph caused by the thrust being cut by a fifth.

9. With the cut-off of the S-IVB's first burn, the vehicle is in orbit..

The 4g reached during boost is not the highest that will be experienced during the mission. Entry through the Earth's atmosphere decelerates the Command Module at about 6½g.]

*http://history.nasa.gov/ap15fj/01launch_to_earth_orbit.htm


Robert h goddard
Robert H. Goddard

  • Robert H Goddard ( 1882 – 1945), was an American physics professor and scientist. He was a pioneer of controlled, liquid-fueled rocketry.

  • In 1926 he launched the world's first liquid-fueled rocket and from 1930 to 1935, he launched rockets that attained higher speeds and higher altitudes.

  • His work in this field was revolutionary, he is recognized as one of the fathers of modern rocketry.

  • In 1914, his first two (of 214) patents were accepted and registered. The first, described a multi-stage rocket. The second, described a rocket fueled with gasoline and liquid nitrous oxide.

  • In 1919, the Smithsonian Institution published Goddard's advanced treatment on rocketry: “A Method of Reaching Extreme Altitudes”. In it he describes his mathematical theories of rocket flight, his experiments with solid-fuel rockets, and the possibilities he saw of exploring the earth's atmosphere and beyond. This book is regarded as one of the pioneering works of the science of rocketry.

  • On Sept. 16, 1959, the U.S. Congress authorized the issue of a gold medal in the honor of Professor Robert H. Goddard


Launching a rocket from earth
Launching a rocket from Earth

  • Earth orbits the Sun at a speed of 107,000 kmh-1 Launching a rocket in the same direction as Earth's movement, will provide a big boost in relative speed (to the Sun) and save a lot of fuel.

  • Earth rotates toward the East, one complete turn each day. This means that at the equator, Earth's surface has a linear speed of about 1600 kmh-1. For this reason most launch sites are located near the equator and rockets are launched toward the east, thus obtaining a 'free' boost from Earth's rotational motion.

  • The eastward launch time is chosen to give the rocket the time to accelerate as it goes partway around Earth.

  • For interplanetary travel, it is important to chose the right time of the year (i.e. the position of the Earth in its orbit) to launch. This is known as a “launch-window”



Rockets can t fly in a vacuum
Rockets can't fly in a vacuum ...

  • On January 12, 1920 a front-page story in The New York Times editorial delighted in heaping scorn on the proposal that travel to the Moon was possible. It expressed disbelief that: “Professor Goddard actually does not know of the relation of action to reaction, and the need to have something better than a vacuum against which to react" !

  • Of course the newspaper was completely wrong (and that is not news). The action-reaction relationship is between the rocket and the gases emitted, and not between the rocket and air.

  • This is an example of how the Law of Conservation of Momentum, which is Newton's third law, can be used to analyse rocket motion.

  • For a rocket at rest, the momentum of the system (rocket + fuel) is zero. According to the Law of Conservation of Momentum this must be preserved.

  • The mass of fuel burned / second is much less than the mass of the rocket, but due to the explosive nature of rocket fuel, the velocity of the hot exhaust gases will be far greater than the velocity of the rocket

  • As fuel is burnt the mass of the rocket diminishes but its velocity increases. The momentum of the system remains zero.


Forces on astronauts
Forces on astronauts

  • Two forces act on an astronaut during launch: the thrust (T) and the astronaut's own weight (mg).

  • Newton’s second law is used to arrive at the relationship between the acceleration of the rocket and the forces on the astronaut,

  • The astronaut experiences this acceleration as an apparent increase in his/her weight of a magnitude that depends on the rocket's acceleration. An acceleration of 4g is experienced as a four-fold increase in weight.

  • Once the rocket reaches a stable orbit a 'weightlessness' is experienced. The astronauts are not really weightless but are continuously falling (in free-fall) toward the centre of the Earth with an acceleration of g (for that altitude)‏

  • During re-entry the spacecraft must be decelerated as quickly and as safely as possible. The acceleration of g downward is opposed by air resistance, which at high speeds can cause the space ship to heat-up to dangerous temperature.

  • The re-entry trajectory is long and shallow to decrease the re-entry g-forces on the astronauts, to decrease the speed of the aircraft and so avoid burn-up or impact at a fatally damaging speed.


Circular orbits
Circular orbits

  • Any object that is moving in a circular path at a uniform speed is accelerating toward the centre of the circular path (its speed is constant, but its direction is constantly changing).‏ The acceleration is called centripetal (centre-pointing) and is associated with a force called the centripetal force:

    Fc = mac (m= mass of the object in circular motion)‏

    For circular motion it can be shown that:

    ac = v2 / r (v = instantaneous tangential speed; r = radius of circle)‏

    Therefore , Fc = mv2 / r

  • For a rock at the end of a string that is being rotated in uniform circular motion, the Fc is the tension in the string. For a satellite or planet in uniform circular orbit, the Fc is the gravitational attraction between the satellite / planet (m) and the central body (M). Hence:

    Fc = Fg

    mv2 / r = G M m / r2

    v2= G M / r

  • 'v' is the orbital velocity of the satellite, it is independent of its mass, but dependent on the mass of the central body (planet or star).


Two types of orbits

Geostationary orbits

A satellite in a geostationary orbit (GO) has an orbital period equal to one sidereal day (23 hours 56 minutes). From the ground, a satellite in GO appears fixed in the sky because its period is the same as the Earth's.

A satellite in GO will have an altitude above the upper van Allen belts of approximately 35 800 km.

An object in GO will need an orbital velocity of about 11,000 km h-1.

Satellites in GO are mainly country- specific communications and weather satellites.

Two types of orbits

Low Earth orbits

  • A low Earth orbit (LEO) is an orbit that lies between the Earth’s upper atmosphere and the lower van Allen radiation belts. The average altitude for a LEO is from 200 km to 1,000 km above the surface of the Earth.

  • An object in LEO will need an orbital velocity of approximately 28.000 km h-1 to maintain the orbit,

  • Objects in LEO will have orbital periods of about 90 minutes.

  • The space shuttle, the Hubble space telescope and the International Space-station are in LEO's


What happened to skylab
What happened to Skylab?

  • Skylab was a space research laboratory constructed by NASA and was launched in 1973.

  • In mid 1979 it re-entered the atmosphere and landed in fiery chunks around the Balladonia Hotel/Motel (Western Australia). There were no injuries.

  • Jimmy Carter, the US President at the time, personally rang the Balladonia Hotel/Motel to apologise to the shaken but not injured staff and guests.

  • What went wrong?

  • Through a process called 'orbital decay' satellites in Low Earth Orbits are slowed down by the friction with the very thin air of the upper atmosphere. This friction results in a loss kinetic energy.

  • The satellite drops down to lower altitude and gains kinetic energy at the expense of gravitational potential energy. It will now be moving faster but a faster velocity means increased friction. At this lower altitude, the air is even denser. The loss in kinetic energy now is even greater, so the process picks up speed and the satellite is soon on a no-return path down to Earth.

  • Most satellites in LEO are small enough to burn up from air-friction before reaching the ground. Skylab was just too big (88,900 kg) and pieces of it managed to hit the ground in WA.

  • The Hubble Space telescope (11,100 kg) orbits in a LEO too and if it is not re-boosted by a shuttle or other means, it will re-enter the Earth's atmosphere sometime between 2010 and 2032.


Keppler s law of periods 3rd
Keppler's Law of Periods (3rd)

  • Notice that,

    • With a system of planets or satellites rotating a central body, the right hand side of the equation remains constant and does not depend on the individual masses of the rotating bodies but only on the mass of the central body

    • If the ratio of the (radius)3 to the (period)2 is known for one body, it will be the same for all the other bodies rotating around the same central star or planet.


Orbital velocity, the gravitational constant, mass of the central body, mass of the satellite and the radius of the orbit

Mass of the satellite and radius of the orbit can be related to the period of the orbit.

Keppler's law can be used to find the relationship between orbital velocity and orbital period.

Orbital velocity is the rectilinear, instantaneous speed tangential to the orbital path in the direction of movement.


Re entry
Re-entry central body, mass of the satellite and the radius of the orbit

  • For spacecraft needing to re-enter the Earth's atmosphere safely, there is only a small angular window available.

  • If the angle is too shallow the spacecraft will bounce on the atmosphere like a flat stone on a pond of water; if the angle is too steep the spacecraft will suffer too much drag from the air and will burn up and the deceleration will be too rapid.

  • The angles differ for different orbits and different spacecraft: for the Apollo capsule the window was between 5o and 7o but for the space shuttle orbiter it is between 1o and 1.5o.

  • There are other consideration for a safe re-entry and a safe landing.

  • Spacecraft in orbit have kinetic energy, as the craft descends even more kinetic energy is produced as it loses gravitational potential energy. On re-entry, some of this energy is converted into a lot of heat by air friction. The effect of this heat can be minimized by 2 methods:

    • By the use of an ablative shield (substances that will burn at a very high temperature and thus use-up the heat, like on the Apollo capsule underside).

    • By using a long and gradual decent with a heat- insulating shield on the underside (the space shuttle uses this method).‏


Re entry1
Re-entry central body, mass of the satellite and the radius of the orbit

  • The heat generated also causes a general radio black-out with the craft as the over-heated air molecules become ionized. For a period of time the craft cannot be contacted and the crew cannot be warned of any dangerous situations that might be developing e.g. being off-course

  • The orbit speeds are in excess of 20 000 km h-1 to decrease from that speed down to a tolerable impact velocity in a distance of just 200 or so km would need very high decelerations (work it out). Hence the re-entry trajectory has too be very shallow resulting in a long path, even then the deceleration is very high and the G-forces on the astronauts can be considerably higher than launch G-forces.

  • Safe impact with the surface is achieved by the use of parachutes deployed at correct altitudes, by splashing in water rather than on hard ground or by gliding down to an airport tarmac like glider (space shuttle)‏


Revision questions 2
Revision questions - 2 central body, mass of the satellite and the radius of the orbit

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

  • Calculate the maximum height, the time of flight and the range of an object with an initial velocity of 30 ms-1 that is projected at 3 different angles: 30o; 45o and 60o

  • Describe a first-hand investigation you performed to analyse data and to calculate initial and final velocity, maximum height reached, range and time of flight of a projectile.

  • Describe Galileo’s contribution to projectile motion.

  • Explain the concept of escape velocity.

  • Outline Newton’s concept of escape velocity.

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

  • 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


Revision questions 3
Revision questions - 3 central body, mass of the satellite and the radius of the orbit

  • Discuss the effect of the Earth‘s motion in space on the launch of a rocket.

  • Describe the changing acceleration of a 2 stage rocket during launch.

  • Describe the forces experienced by astronauts during launch.

  • Describe the forces involved with uniform circular motion.

  • Compare low Earth and Geostationary orbits

  • Account for the orbital decay of satellites in low Earth orbit.

  • Calculate the orbital speed and period of a satellite orbiting at an altitude of 350 km above the Earth's surface. Explain why this is not a Geostationary orbit.

  • The average radius of Earth's orbit is 149,600,000 km and its period of rotation is 365.26 days. If Jupiter rotates the Sun in 4332.71 Earth days, calculate the average distance of Jupiter from the Sun

  • Define the term orbital velocity and using Keppler's 3rd law find its relationship to the period of the orbit

  • Outline issues that relate to a safe re-entry and touch down for a manned spacecraft in parking orbit around the Earth. Identify which of these would not apply to a safe landing on the Moon's surface


3 gravity
3. Gravity central body, mass of the satellite and the radius of the orbit


Gravity
Gravity central body, mass of the satellite and the radius of the orbit

  • Gravity is the force of attraction between two or more masses.

  • Each mass is thought of having an infinite gravitational field surrounding it in space

  • The gravitational force exerted by an object is proportional to its mass.

  • The strength of the field obeys the inverse- square law (it diminishes as the inverse of the square of the distance from the centre of mass).

  • Gravity is the weakest of the four known forces, but large bodies like the Sun and the planets exert strong gravitational fields because of their large masses

  • Gravity holds the planets and satellites in their orbits it even stops the thermonuclear fusion reaction in the Sun from exploding it apart.

  • Gravity shapes the structure and evolution of stars, galaxies, and the entire universe...


Newton s law
Newton's Law central body, mass of the satellite and the radius of the orbit

Newton's Law of Universal gravitation states”

“Every object in the universe attracts every other object with a force directed along the line joining the centres of mass of the two objects.

The force point toward each object and is proportional to the product of their masses and inversely proportional to the distance separating their centres”.

G = 6.67300 × 10-11 m3 kg-1 s-2 (Universal gravitation constant)‏


Newton s law and the motion of satellites
Newton's law and the motion of satellites central body, mass of the satellite and the radius of the orbit

  • As previously shown; Newton's Law of gravitation is needed to derive an expression for the orbital velocity of a satellite in near circular orbit (it is also used for elliptical orbits but the math involved is beyond our scope)‏.

  • Newton's Law can also be used to derive Keppler's third law.

  • With these two laws the path or orbit of any satellite or planet can be analyzed and calculated. Predictions can be made as to its future positions to the extent that spacecraft can find them in the vastness of space.


Slingshot
Slingshot central body, mass of the satellite and the radius of the orbit

  • The slingshot effect is also know as a 'flyby' or as a 'gravity assist' trajectory. It consists of piloting or directing the spacecraft close to a planet in order to achieve a change in the direction or a change in velocity without the expenditure of fuel.

  • Depending on the direction of approach of the spacecraft relative to the planet's motion, a spacecraft' s velocity can be increased (if the destination is further on) or decreased (if a parking orbit is the aim)

  • In a gravity-assist trajectory, momentum is transferred from the orbiting planet to a spacecraft approaching it (it's not unlike an elastic collision between two billiard balls, except one of the balls is millions of times more massive than the other!)‏

  • Both gain/lose the same amount of momentum but the planet shows negligible change in its velocity as it is so massive, whereas the spacecraft shows a significant change in it's speed (and / or direction)‏

  • Voyager 2 in its flyby with Jupiter increased its speed from 10 kms-1 to almost 30 kms-1 without burning any fuel!


Variations in gravitational force
Variations in gravitational force central body, mass of the satellite and the radius of the orbit

  • Gravitational force depends on the mass and is inversely proportional to distance.

  • A body which has twice the mass of another will have twice the gravitational force at the same distance

  • Doubling the distance from a mass will reduce its gravitational force to a quarter

  • The value of g on a planet's surface with the same radius as Earth's but ½ the mass will be 9.8 / 2 = 4.9 ms-2

  • The value of g on a planet's surface with the same mass as Earth's but ½ the radius will be 4 x 9.8 = 39.2 ms-2

  • On a local level, because g depends on mass, the denser rocks will exert a higher gravitational force.

  • Satellites can measure the surface variation of g for the Earth and have produced colourful maps. They show that where the crust is thinnest, the ocean floor, g is smallest.


Revision questions 4
Revision questions - 4 central body, mass of the satellite and the radius of the orbit

  • Define Newton’s Law of Universal Gravitation.

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

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

  • Discuss the factors that affect the strength of the gravitational force.

  • The International Space Station presently has a total mass of approximately 227,000 kg and is in a LEO at an average altitude of 350 km. Calculate:

    • The value of g at this altitude

    • The orbital velocity of the ISS

    • Its period of rotation

    • The work done to place it into orbit

    • Its kinetic energy

    • The r3 / T2 ratio for the ISS and hence the period of rotation of the Moon. (Moon – Earth distance = 3 x 10 5 km)‏

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


4. “c” central body, mass of the satellite and the radius of the orbit


To aether or not to aether
To aether or not to aether? central body, mass of the satellite and the radius of the orbit

  • In 1801 Thomas Young performed the double-slit experiment and proved the wave nature of light. All waves known at that time needed a medium in which to propagate, e.g. sound cannot travel in a vacuum. But, no medium could be found for the propagation of light and light was known to travel in a vacuum.

  • The luminiferous aether or ether was hypothesized as being the medium that pervaded the entire universe (even solid objects) and that it was the medium through which all electromagnetic waves propagated.

  • The aether was given a set of properties:

    • The aether should: fill all of space and be stationary in space; be perfectly transparent; permeate all matter; have a low density and have great elasticity, it must be very light but almost incompressible to allow light to travel so fast.

    • Yet, it must allow solid bodies to pass through it freely, without any resistance, or the planets would be slowing down.

  • It is the relative motion of the Earth through the ether that Michelson and Morley tried to measure.


M m detecting the aether wind
M & M - Detecting the Aether Wind central body, mass of the satellite and the radius of the orbit

  • The thought behind the Michelson-Morley (M&M) experiment is better understood by considering these two sketches of two swimmers (a similar drawing was included in their original paper)‏

  • Swimmers A and B both swim at exactly the same speed of 5 m/s. They both start and return at point X.

  • In sketch 1 (no current)- both start and return at the same time.

  • In sketch 2 (a 3 m/s current)

    • Swimmer A takes 12.5s to swim to Y (@ 8 m/s) and 50s to swim back to X (@ 2 m/s). A total of 62.5s

    • Swimmer B has to take an angled course to point P (because of the current) in order to return to the same point. The course is longer (125 m) and it takes him 25s for both trips a total of 50s

  • The river represents the aether; the two swimmers represent Earth moving along and across the aether. The movement of the solar system through the aether is the current.

  • M&M designed equipment using an interferometer that was calculated to be up to 40 times more accurate than was needed.


M m s experiment
M & M 's experiment central body, mass of the satellite and the radius of the orbit

  • The paths A and B are represented by the swimmers A and B in the preceding example. As the Experiment is rotated their respective arrival times at the interferometer screen should change in the presence of an aether current. This change will be easily detected by a shift in the interference fringes produced.

  • Even though the experiment was repeated many times, at different times of the year (position in Earth's orbit) and at different altitudes.NO SIGNIFICANT CHANGE WAS EVER DETECTED!

A null result, conclusion: NO AETHER ??


M m s null result
M&M' s central body, mass of the satellite and the radius of the orbitnull result

  • Albert Michelson and Edward Morley were experienced and rigorous scientists (Michelson had just succeeded in making the most accurate measurement of the speed of light of the time – a figure that is accurate to this day).

  • Their experiment was well planned, well constructed and provided reproducible results. They were not satisfied in repeating it a few times but actually repeated it many times in different locations and at different times during the year.

  • Both were firm believers in the aether theory, so if there was any bias at all, it would have been for a positive result, not a null one. Both were very reluctant in reaching the only conclusion possible from their results: that the existence of the aether was severely in doubt. Both received the Nobel prize for this fundamental “ null” experiment.

  • Many other physicists over the years (to this day!) have had trouble in letting the aether theory go and many modifications of the aether theory have been tested, but to date each test has failed.

  • Some years after the Michelson-Morley experiments, Albert Einstein proposed his theory of special relativity. Even though Einstein did not set out to prove that the aether did not exist, a result of his theory was that the aether is not needed.

  • In contrast to the aether theory, Einstein's special theory of relativity has had confirmation from many of experiments.


Inertial frames of reference
Inertial frames of reference. central body, mass of the satellite and the radius of the orbit

  • An inertial frame of reference is one that is at rest or that is moving at constant velocity

  • A non-inertial frame of reference is one that is accelerating.

  • By definition all inertial frames of reference are identical – all will produce the same results to experiments carried out in them

  • Also by definition any mechanical experiment carried out in an inertial frame of reference will provide NO information regarding the state of uniform motion or rest of the frame.

  • i.e. You cannot tell whether you are moving or not unless you look at another inertial frame of reference and then you can measure your relative speeds.

  • This is known as Galilean Relativity. Galileo was the first to recognize that any observation within the inertial frame of reference would not help to find out whether the frame of reference was at rest or moving at a steady velocity.

  • Einstein took Galilean Relativity a step forward, by including electromagnetism. He proposed that any electromagnetic experiment would also give the same results in any inertial frame of reference.


Inertial or non inertial
Inertial or non-inertial? central body, mass of the satellite and the radius of the orbit

  • A ballistic cart is good piece of equipment to demonstrate the difference between the two frames of reference:

    • In the first two experiments the ball propelled upward by the vertical “cannon” returns cleanly to the cannon. An imaginary observer on the cart cannot tell by just looking at the ball whether he is in 1 or 2 (assuming he cannot see outside the train)

    • In the third experiment the cart is accelerating (pulled by a string) and the ball is “left behind” the imaginary observer knows he is accelerating.


Einstein s theory of special relativity
Einstein's Theory of Special Relativity central body, mass of the satellite and the radius of the orbit

  • In 1864 James Maxwell derived a set equations that described the properties of electric and magnetic fields and how they interact and give rise to each other. Using his equations he was able to calculate the speed of light as being 3×108 ms-1 (c ) purely as a mathematical consequence of the interaction of the two fields in a vacuum.

  • In 1905 Albert Einstein proposed that the laws of inertial frames must also apply to Maxwell's equations just as they applied to the rest of physical laws.

  • Einstein's Special Relativity states that: “the Laws of Physics are the same in any inertial frame and that includes any measurement of the speed of light - in any inertial frame the speed of light will always be c (A consequence of special relativity is that there is no natural rest-frames in the universe. Hence, not only is the aether not necessary, but also it cannot exist as it would provide a natural frame at rest.)‏

  • The speed of light will be constant for all observers in any inertial frame of reference.


Einstein s thought experiments
Einstein's thought experiments central body, mass of the satellite and the radius of the orbit

  • Einstein formulated his theory of relativity using thought experiments. He did this to investigate situations, that at the time, could not be tested in reality.

  • Most of Einstein's deductions using his thought experiments have since received experimental confirmation. His thought experiments were based on a deep understanding of physics and on his natural genius.

  • His famous thought experiment that led him to conclude that the aether model did not work for light is simplified and illustrated here ----->

  • The result predicted by the aether model is that he will not see his reflection. Since light can only travel at C through the aether. The light from the man's face cannot catch up to the mirror to return as a reflection.

  • This violates the principle of inertial frames (in an inertial frame of reference you cannot perform any experiment to tell that you are moving). Therefore, he must be able to see his reflection

  • Einstein concluded that: the reflection will be seen as normal so that the principle of inertial frames would not be violated; he described the aether model as superfluous.

A man is sitting in a train facing forwards. The train is moving at the speed of light. He has a mirror in front of him; will he be able to see his reflection in the mirror?


Spacetime
Spacetime central body, mass of the satellite and the radius of the orbit

  • Einstein's prediction that the speed of light is a constant to all observers (at rest or moving uniformly) led to a number of logical but startling consequences.

  • It changed how we view time and space. The speed of light was constant but space, time and even mass were relative to the observer. The term “spacetime” was coined with this new understanding

  • Spacetime is viewed as a consequence of Einstein's 1905 Theory of Special Relativity, but is treated more fully in Einstein's next corner- stone of modern physics: the Theory of General Relativity.

  • Special relativity gave birth to the concepts of time dilation, length contraction, mass dilation and mass/energy equivalence. (it was only one, of four paradigm-changing papers Einstein published in 1905 at the age of 26!)‏

  • For many years there was no evidence for Einstein's Theory of Special Relativity. However, starting with the confirmation of his prediction that the light from a distant star would bend when passing close to the Sun, evidence supporting the theory was not long in coming.


Relativity of simultaneity

Observer central body, mass of the satellite and the radius of the orbitB is in another inertial frame, he sees the train passing and the light turn on . But the light going to the front door takes a little longer to reach the photocell because the front of the train has moved a distance 'vt' forward. i.e. (0.5L + vt) / C

Vice versa, the light going to the back of the train takes a little less time. i.e.(0.5L - vt) / C

For B, the doors open at different times.

Note: The value of C is the same for both observers.

Relativity of simultaneity

Adapted from Einstein's thought experiment:

  • A train of length L has a door at both ends. They are operated by means of a photocell.

  • When the light from a source in the middle of the train hits the photocells the doors open

  • An observer A standing in the middle of the train can turn the light on. When the train reaches a uniform speed 'v' and when he passes a station he turns the light on.

  • An other observer B is standing stationary at the station as the train passes.

  • For A,the doors open at the same time.

  • He is in an inertial frame and light must reach the photocells at the same time. (0.5L / C)‏

Simultaneity is relative to the observer


Time dilation
Time dilation central body, mass of the satellite and the radius of the orbit

  • Each will see their own clock as

Adapted from Einstein's thought experiment;

  • For this experiment we will use what can be named Einstein's light clock:

  • But each will see the other person's clock as:

  • A flash of light bounces back and forth between two mirrors, d metres apart. Each time it hits the top mirror the clock advances one click.

  • Time between clicks = 2d / C

  • Observer A is in a train moving at a constant velocity 'v' past a railway station. A stationary observer B is at the station.

  • Both observers have identical light clocks. In each case they see their own light clock as stationary (with respect to themselves) and the other person's clock as a moving clock at velocity v

  • Using Pythagoras and substitution, it can be shown that :


Experimental evidence for time dilation
Experimental evidence for time dilation central body, mass of the satellite and the radius of the orbit

  • One of the implications of time dilation is that clocks moving at relativistic speeds (speeds near the speed of light) will appear to run significantly more slowly to an observer in another frame of reference.

  • This was confirmed in 1963 (40 years after Einstein proposed it) when the unstable subatomic particles called Muons were measured to have longer half-lives at near light speeds. The increase was exactly as predicted by the theory.

  • As technology has improved, so has our ability to measure fractions of seconds more and more accurately. This increased accuracy has allowed the theory to be confirmed by comparing identical clocks at rest with ones in fast moving jets. They too confirmed Einstein's formula.

  • The demand for more and more accurate GPS' has resulted in having to correct, using the time dilation formula, for time dilation effects due to the orbital speeds of the satellites in the system.

  • Astronomical data from receding supernovae agrees with time dilation calculations.


Length contraction
Length contraction central body, mass of the satellite and the radius of the orbit

  • The length of any object in a moving frame will appear foreshortened in the direction of motion, or contracted. The length is maximum in the frame in which the object is at rest.

  • The length of an object measured within its inertial frame is called its proper length Lo. The length of an object in a frame on relative motion to the observer's frame is Lv

  • Length contraction (also known as Fitzgerald Contraction), like time dilation is symmetrical and is relative to the observer.


Evidence for length contraction
Evidence for Length contraction central body, mass of the satellite and the radius of the orbit

  • Length contraction and time dilation are related as both are connected to speed. This is best illustrated with the Muon evidence for time dilation

  • The increase in the Muon's half-life is seen from the perspective of an observer on Earth as an example of time dilation.

  • But from the Muon's point of view its half-life is a constant and is not changed by its speed. What has changed is the distance outside its frame of reference that it must cover in that half-life.

  • The Muon "sees” that distance foreshortened by length contraction giving it time to cover it in its short life span.


Atomic clocks and the metre
Atomic clocks and the metre. central body, mass of the satellite and the radius of the orbit

  • In 1793 the French defined the unit of length to be the metre as 1 ten millionth of the Earth's quadrant passing through Paris.

  • For many years this standard was used as the reference metre – the distance between two fine lines on a platinum-iridium bar kept at a standard temperature

  • Today the metre is defined using the speed of light and using very small fractions of a second measured by atomic clocks:

  • “The meter is the length of the path traveled by light in a vacuum during a time interval of 1/C seconds. Where C is the speed of light, taken as:

    C = 299,792,458 ms-1

  • “A second is defined as the duration of 9,192,631,770 cycles of microwave light absorbed or emitted by the hyperfine transition of cesium-133 atoms in their ground state undisturbed by external fields.”


Mass energy equivalence
Mass – Energy equivalence central body, mass of the satellite and the radius of the orbit

  • The mass–energy equivalence is the concept that mass is energy, and that energy is mass.

  • In special relativity this relationship is expressed using the mass–energy equivalence formula

    E = mC2

    (Where: E = energy; m = mass and C = the speed of light in a vacuum)‏

  • The formula (voted the best known formula of all time) was derived by Albert Einstein, in 1905 in the paper "Does the inertia of a body depend upon its energy-content?"

  • The concept of mass–energy equivalence unites the concepts of conservation of mass and conservation of energy.

  • The mass–energy equivalence formula was used in the development of nuclear reactors and of the atomic bomb. During Fission (or Fusion) part of the mass is converted to energy. 1 kg of mass is equivalent to 9 x 1016 Joules!

  • The Sun converts 4.26 million metric tons of mass to energy every second

  • Mass-energy equivalence led Einstein to the concept of mass dilation


Mass dilation
Mass dilation central body, mass of the satellite and the radius of the orbit

  • The term mass in special relativity usually refers to the rest-mass of the object, which is the Newtonian mass as measured by an observer with the same velocity as the object. The invariant mass is another name for the rest-mass of single particles.

  • The term relativistic mass includes a contribution from the kinetic energy of the body, and increases as the object's velocity increases..

    (Where: mv= relativistic mass; m0 = rest mass; v is the velocity relative to the observer and c = speed of light)‏

  • Mass dilation means that a particle moving at relativistic speeds will have a greater mass than its rest mass when measured by an observer in a different frame of reference.


Evidence for mass dilation
Evidence for Mass dilation central body, mass of the satellite and the radius of the orbit

  • Since the birth of nuclear energy the evidence for the mass-equivalence concept and its derivative the mass dilation equation has been substantial.

  • Particle accelerators daily accelerate subatomic nuclei to relativistic speed and measure their masses using the mass dilation relationship to achieve required momenta for their collision experiments

  • Nuclear reactors produce energy by (mostly) uranium fission, the energy out put can be calculated using the mass-energy equivalence formula

  • Both formulae are important in astronomy when considering stellar evolution and receding star systems

  • Every particle in the universe is in motion and hence has a relativistic mass.


Deep space travel
Deep-Space travel central body, mass of the satellite and the radius of the orbit

  • A light-year (ly) is a measure of distance. It is the distance traveled by light in one year in vacuum at the speed of 3 x 10 8 m. It is a very long distance ( 9.4605284 × 1015 meters)‏

  • The closest star to the solar system is Proxima Centauri, it is 4.2 ly away.

  • Even traveling near light speed a space ship to Proxima Centauri would have sustain its crew for a long voyage.

  • Special relativity would help in some ways and hinder us in others. Moving at say 0.75C would shorten our path considerably because of length contraction.

  • And we would return to Earth in less ship years than had passed on Earth, because of time dilation.

  • But mass dilation would make it almost impossible for us to reach relativistic speeds as it would require increasing and vast amounts of energy to accelerate to those speeds.

  • Our Galaxy, just one of billions in space, it is 100,000 ly across. Intra-Galactic travel will need the discovery of other means of travel: hyperspace, wormholes or something that has not even been thought of.

  • NASA projects that we may be able to get to Mars by 2030. Mars is “just” 11 light-minutes away.


Revision questions 5
Revision questions - 5 central body, mass of the satellite and the radius of the orbit

  • Outline the features of the aether model.

  • Describe and evaluate the Michelson-Morley experiment.

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

  • Interpret the results of the Michelson-Morley experiment

  • Outline the nature of inertial frames of reference

  • Describe an investigation you performed to distinguish between non-inertial and inertial frames of reference

  • Discuss the principle of relativity

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

  • Identify the implications of C being a constant to all observers.


Revision questions 6
Revision questions - 6 central body, mass of the satellite and the radius of the orbit

  • Explain qualitatively and quantitatively the consequence of the relativity of simultaneity.

  • Explain qualitatively and quantitatively the consequence of the equivalence between mass and energy.

  • Explain qualitatively and quantitatively the consequence of length contraction.

  • Explain qualitatively and quantitatively the consequence of mass dilation.

  • Explain qualitatively and quantitatively the consequence of time dilation.

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

  • Discuss the relationship between theory and the evidence supporting it, using Einstein’s predictions based on relativity that were made many years before evidence was available to support it.

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

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


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