SPH 4U Unit #1 Dynamics Topic #13: Universal Gravitation and Satellites (Teacher)

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1.13.1 The Development of the Theory of Universal Gravitation

Before Newton developed the theory of universal gravitation, there were two separatenotions of gravity. Terrestrial gravity was thought to be the force holding things to thesurface of the Earth. It was not clear that this force was the same as astronomicalgravity which was responsible for the motion of the planets, moons and comets.

Newton showed that these two forces were actually the same; he unified terrestrial andastronomical gravity.

There is a popular story that Newton was sitting under an apple tree, an apple fell onhis head, and he suddenly came up with the theory of universal gravitation. This isalmost certainly not true.

What is thought to have actually happened is that Newton, upon observing an apple fallfrom an apple tree, thought: “if the apple accelerates toward the ground, there must bea force that acts on it. Let's call this force "the force due to gravity", and its accelerationwe’ll call “the acceleration due to gravity". Also, if the apple tree was twice as high, theapple would still accelerate toward the ground, so the force due to gravity must reach tothe top of the tallest apple tree. Perhaps, the force of gravity reaches even further, perhaps all the way to the Moon! This would mean that the orbit of the Moon around theEarth is caused by the force of gravity.

SPH 4U Unit #1 Dynamics Topic #13: Universal Gravitation and Satellites (Teacher)

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This idea can be illustrated with the thought experiment that follows. If we fire acannonball horizontally from a tall building, it will eventually fall to earth because of theforce due to gravity, as indicated by the trajectory marked 1 in the figure. As weincrease the muzzle velocity of the cannon, the projectile will travel further and furtherbefore returning to earth. Finally, Newton reasoned that if the cannon projected thecannonball with exactly the right velocity, the cannonball would travel completely aroundthe Earth, always falling in the gravitational field but never reaching the Earth. That is,the cannonball is put into orbit around the Earth. Newton concluded that the orbit of theMoon was of exactly the same nature: the Moon continuously "fell" in its path aroundthe Earth because of the acceleration due to gravity, thus producing its orbit.

By such reasoning, Newton concluded that terrestrial and astronomical gravity were infact the same and that any two objects in the Universe exert gravitational attraction oneach other, with the force having a universal form:

(1.11)

Note that G is the same everywhere hence the name “Newton’s Law of UniversalGravitation”

SPH 4U Unit #1 Dynamics Topic #13: Universal Gravitation and Satellites (Teacher)

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1.13.2 Weight and the Gravitational Force

In everyday conversation, the terms mass and weight are often used to mean the samething. In reality they are quite different. Weight is defined as: the gravitational forceexerted on an object of a certain mass by another mass. Mathematically, the weight is:

(1.11)

where one of the masses is the mass of the Earth. We can also define the accelerationdue to gravity as:

The acceleration due to gravity is approximately the product of the universalgravitational constant G and the mass of the Earth M, divided by the radius of the Earth,r, squared. (We assume the Earth to be spherical and neglect the radius of the objectrelative to the radius of the Earth in this discussion.) The measured gravitationalacceleration at the Earth's surface is found to be about 9.80 m/second/second.

So the weight of a mass m at the surface of the Earth is obtained by multiplying the

mass m by the acceleration due to gravity, g, at the surface of the Earth.

This means that the mass of a specific object is a constant for that object but its weightdepends on its location. If we transport an object of mass m to the surface of the Moon,the gravitational acceleration would change because the radius and mass of the Moonboth differ from those of the Earth.

SPH 4U Unit #1 Dynamics Topic #13: Universal Gravitation and Satellites (Teacher)

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Cavendish, Henry(1731-1810)

1.13.3 Using Universal Gravitation to Weigh the Earth

Towards the latter half of the 18th century, spurred by hisinterest in the structure and composition of the interior of theearth, Henry Cavendish in a 1783 letter to his friend Rev.John Michell discussed the possibility of devising anexperiment to "weigh the earth." Borrowing an idea from the French scientist Coulomb who had investigated the electricalforce between small charged metal spheres, Michellsuggested using a torsion balance to detect the tinygravitational attraction between metal spheres and set aboutconstructing an appropriate apparatus. He died in 1793,however, before conducting experiments with the apparatus.

The apparatus eventually made its way to Cavendish'shome/laboratory, where he rebuilt most of it. His balance was constructed from a 2metre wooden rod suspended by a metal fiber, with 5 cm. diameter lead spheresmounted on each end of the rod. These were attracted to 160kg. lead spheres broughtclose to the enclosure housing the rod, roughly as shown below. He began hisexperiments to "weigh the world" in 1797 at the age of 67, and published his result in1798 that the average density of the earth is 5.48 times that of water. His work wasdone with such care that this value was not improved upon for over a century. Themodern value for the mean density of the earth is 5.52 times the density of water.Cavendish's extraordinary attention to detail and to the quantification of the errors inthis experiment has lead C. W. F. Everitt to describe this experiment as the firstexperiment in modern physics.

SPH 4U Unit #1 Dynamics Topic #13: Universal Gravitation and Satellites (Teacher)

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1.13.4 Apparent Weight

In our daily experience, our apparent weight is equal to the Normal force applied by thefloor upwards on our feet (This is what a bathroom scale measures). When you ride inan elevator, the normal force you feel varies which you interpret as your apparentweight. Three situations are worth examining here:

1. The elevator is at rest or moving at a constant speed

In this case, the normal force you feel is exactlyequal to your weight and you feel “normal”

2. The elevator accelerates upward

The normal force in this case is greater than itusually is so you feel “heavier”

3. The elevator accelerates downward

The normal force in this case is less than itusually is so you feel “lighter”

4. The cable snaps

The normal force in this case is zero. Anysituation in which we feel no normal (supporting) force,we feel as though our weight were zero, aka“weightless”. This sensation has also been called “zerogravity” and “micro gravity” however these terms aremisleading since the force due to gravity is not zero.

SPH 4U Unit #1 Dynamics Topic #13: Universal Gravitation and Satellites (Teacher)

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Eg.#1 Determine the apparent weight of a 50 kg. person standing in an elevator,

a) that is motionless b) that accelerates upward at 2.0 m/s2

c) that moves downward at 1.5 m/s d) That is in free-fall

1.13.5 Satellites and Space Stations in Geosynchronous Orbits

A satellite in geosynchronous orbit circles the earth once each day. In other words, itsorbital period is 24 hours.

Eg.#2 What distance above the Earth’s surface must a satellite be in order to begeosynchronous (the radius of the Earth is 6378 km.)?

SPH 4U Unit #1 Dynamics Topic #13: Universal Gravitation and Satellites (Teacher)

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Inertial Upper Stage – Boeing corp.

The Geometry of a Hohmann Transfer

For a satellite's orbit period to be one Earth day, it must be approximately 36000kilometers above the earth's surface. That is a lot higher than the Shuttle ever goes(usually about 300 kilometers). We calculate this height using, what are today, commongeometric formulas.

To stay over the same spot on earth, a geostationary satellite also has to be directlyabove the equator. Otherwise, from the earth the satellite would appear to move in anorth-south line every day. We call that "orbiting in the equatorial plane."

The Shuttle's orbit is alwaysinclined to the equator by atleast 28.5 degrees. Given thisand the Shuttle's relatively loworbit, getting a satellite from itsdeployment orbit to its finalgeosynchronous orbit takes anInertial Upper Stage (IUS) for aboost, and something called aHohmann transfer.

A Hohmann transfer is a fuelefficient way to transfer from onecircular orbit to another circularorbit that is in the same plane(same inclination), but a differentaltitude.

To change from a lower orbit (A) to a higher orbit(C), an engine is first fired in the opposite directionfrom the direction the vehicle is traveling. This willadd velocity to the vehicle causing its orbit tobecome elliptical (B). This elliptic orbit is carefullydesigned to reach the desired final altitude of thehigher orbit (C). In this way the elliptic orbit ortransfer orbit is tangent to both the original orbit(A) and the final orbit (C). This is why a Hohmanntransfer is fuel efficient. When the target altitude isreached the engine is fired in the same manner asbefore but this time the added velocity is plannedsuch that the elliptic transfer orbit is circularized atthe new altitude of orbit (C).

SPH 4U Unit #1 Dynamics Topic #13: Universal Gravitation and Satellites (Teacher)

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The Three Parts of the G.P.S. System

1.13.6 Global Positioning System

GPS, which stands for Global Positioning System, is the only system today able toshow you your exact position on the Earth anytime, in any weather, anywhere to withinabout 10 metres. Even greater accuracy, usually within less than one metre, can beobtained with corrections calculated by a GPS receiver at a known fixed location.

GPS has 3 parts: the space segment (the satellites), the user segment (the handheld or car mounted receiver), and the control segment (5 ground stations to ensure the satellites are working properly).

SPH 4U Unit #1 Dynamics Topic #13: Universal Gravitation and Satellites (Teacher)

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GPS Nominal ConstellationTwenty Four Satellites in Six Orbital Planes

Four Satellites in Each Orbital Plane

1.13.6a GPS Satellites

The complete GPS space system includes 24 satellites, 20,000 km. above the Earth,which each take 12 hours to orbit the Earth. They are positioned so that signals from sixof them can be received nearly 100 percent of the time at any point on Earth. Sixsignals are required in order to get the best position information. G.P.S. satellites areequipped with very precise clocks that keep accurate time to within three nano-seconds. This precise timing is important because the receiver must determine exactlyhow long it takes for signals to travel from each GPS satellite. The receiver uses thisinformation to calculate its position.

The first GPS satellite was launched in 1978. The first 10 satellites were developmentalsatellites, called Block I. From 1989 to 1993, 23 production satellites, called Block II,were launched. The launch of the 24th satellite in 1994 completed the system.

SPH 4U Unit #1 Dynamics Topic #13: Universal Gravitation and Satellites (Teacher)

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1.13.6b GPS Ground Control Stations

The GPS ground control segment consists of five unmanned monitor stations locatedaround the world:

Hawaii in the Pacific OceanKwajalein in the Pacific Ocean; Diego Garcia in the Indian Ocean; Ascension Island in the Atlantic Ocean; and Schriever Air Force Base in Colorado Springs, Colorado (the master station)

These ground stations broadcast signals to the satellites and also track and monitorthem

1.13.6c GPS Receivers

GPS receivers detect, decode, and process GPS satellite signals. Many differentreceiver models are already in use. The typical hand-held receiver is about the size of acellular telephone, and the newer models are even smaller.

SPH 4U Unit #1 Dynamics Topic #13: Universal Gravitation and Satellites (Teacher)

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Worksheet 1.13

1. Four masses are located on a plane as shown below. What is the net gravitational

1force on m due to the other three masses?

2. Do you agree with the statement, “There is no location anywhere in the universewhere a body can exist with no force acting on it”? Explain your answer.

Given that , there can never be a location where since this would never

happen unless .

SPH 4U Unit #1 Dynamics Topic #13: Universal Gravitation and Satellites (Teacher)

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3. At a certain point between earth and the moon, the net gravitational force exerted onan object by Earth and the Moon is zero. The Earth – Moon centre to centre distance is384000 km. and the mass of the moon is 1.2% the mass of the Earth

a) Where is this point located?

b) What is the physical meaning of the root of the quadratic equation whose valueexceeds the Earth–Moon distance?

This is the location where the forces are equal but are in the same direction.

SPH 4U Unit #1 Dynamics Topic #13: Universal Gravitation and Satellites (Teacher)

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4. The International Space Station follows an orbit that is, on average, 450 km. abovethe surface of the Earth.

a) Determine the speed of the ISS

b) Determine the orbital period of the ISS

SPH 4U Unit #1 Dynamics Topic #13: Universal Gravitation and Satellites (Teacher)

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5. Astronomers have identified a black hole at the centre of the galaxy M87. From theproperties of the light observed, they have measured material orbiting at a distance of5.7 x 10 m. from centre of the black hole, travelling at an estimated speed of 7.5 x17

10 m./s.5

a) Assuming that the material is in a circular orbit, calculate the mass of the black hole.

b) What is the ratio of the mass of the black hole to the mass of the Sun (1.99 x 1030

kg.)