LEARNING OBJECTIVES:

I will be able to apply Newton's theory of gravitation to orbits during today's class.

WORDS O' THE DAY:

• Centripetal ("towards the center")
• Centrifugal ("away from center")
• Gravitational Constant: G = 6.674 × 10^-11 Nm²/kg²
• Period ("Time for one revolution")

FORMULAE OBJECTUS:

• v²/r: centripetal acceleration
• mv²/r: centripetal force
• F_g = Gm₁m₂/r²: This is Newton's famous equation for gravitational attraction.

Back in the day, Sir Isaac discovered that there was a profound relationship between two object that depended on the mass of each object and the distance between them.

However, while developing the math, he found that he was CONSTANTLY off by a specific amount. In otherwords he found a proportional relationship thusly:

F_g was proportional to m₁m₂/r²

Through very careful measurements over the last 400 years, we have very precisely determined the amount that must be multiplied to make those two terms EQUAL to each other... we call that term G.

G = 6.674 × 10^-11 Nm²/kg²

The gravitational force between objects is found by multiplying the mass of each object by the "G" the gravitational constant divided by the square of the distance between those two masses in meters (square). Oddly enough, gravity is a very, very weak force. A simple bit of friction here on Earth causes objects to NOT be drawn together....we'll discuss at length

WORK O' THE DAY:

The formula for determining the acceleration associated with an object moving in a circular motion is:

v²/r

With that in mind please determine/derive a new form of Newton's 2nd Law that will allow us to calculate the force required to keep an object moving in circular motion.

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Let's review several situations we've already discussed, but now we'll do so *quantitatively* (ie, eg, for example, therefore WITH MATH):

• What is the source of the force required to keep a satellite moving in a circular orbit around the Earth?

Now let's work with the idea of finding the velocity of an object moving in circular motion:

1) What information do we need to calculate the speed at which the ISS (International Space Station) moves in orbit around the Earth?

• The period (T) of the ISS
• The distance from the center of the Earth (radius)
• That should do it! Work with your team to research those values and calculate the rotational velocity of the ISS.
• Why do you suppose that term is sometimes referred to as the tangential velocity? Please discuss!

2) What is the source of the force required to keep an object tethered to the end of a string moving in circular motion and can we determine the centripetal force required to keep that object moving in a circle?

• • Please grab some string and a washer

  • CAREFULLY -- see if you can move the washer in a slow enough motion so that you can count the number of revolutions per second

  • Determine the other values you need to calculate F_c.

  • Do it!

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  We'll stop here today

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3) Now let's try a more advanced situation: Imagine you are in (on?) the Seattle Great Wheel:

What is the source of the force acting on you to keep you moving in a circle at constant rate?

Hmmm, here's an interesting challenge for you and your team: where is your weight greater: At the top of the ride or at the bottom of the ride.

Why?

Determining the force you experience on the wheel takes a bit of work. Let's do that now:

The Wheel does one full revolution about every 4.0 minutes, calculate the period (T) of that motion.

The diameter of the wheel is 175 feet, calculate the circumference of the wheel.

Now calculate the rotational speed of the wheel (in m/s if you please!)

Now calculate the centripetal force you experience in your motion around the center of the wheel (mv²/r) if we assume your mass is 75 kg.

Now let's get a wee bit nasty:

Your weight at the top of the ferris wheel is not the same as your weight at the bottom of the ferris wheel (it's true! although the Great Seattle Wheel moves so slowly the difference is rather slight)

Let's start with your weight (we'll deal with sig figs at the end of our work as per normal):

W = mg = (75 kg)(9.81) = 735.75 N

Do a sketch of the situation and identify the forces acting on you at the top of the wheel and the bottom of the wheel (remember sum-of-the-forces????)

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Newton's Theory of Gravity goes beyond his famous "3 Laws". Through careful/detailed study Newton was able to determine the mathematical relationship between the mass of objects and the gravitational force exerted by those objects please commit the Universal Gravitational Constant to memory ASAP:

6.674 × 10^-11 Nm²/kg²

To wit:

F_g = Gm₁m₂/r²

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Please find the gravitational force between the Earth and the ISS (You'll need to find specific values...)

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An empty 747-400 (m = 250,000 kg) and a medium sized cargo ship (mass = 350,000 kg) are floating in deep space The two objects are 2.5 m apart.

 

What is the significance of the 'weasel words' "Deep Space"?

Work with your team to analyze that situation. Will the gravitational force between those two objects be substantial, weak, or somewhere inbetween?

Define what you mean by substantial and weak and try again!

What is the magnitude of the the gravitational force between them?

How fast will the the 747 accelerate?

What does this exercise tell us about the strength of the gravittional force?

 

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Problem Sets:

Step One: Read the problem quietly to yourself. The person sitting most closely to the nw corner of your table will assign duties as to quantities to look up and conversions to be made.

No one else can initiate conversation although they can ask clarifying questions if they are touching the talking object.

Problem #0:

Bad news... your buddies forgot you were out playing space hockey and flew off in your spaceship leaving you marooned 1500 meters above the surface of the asteroid Vesta. You are sorta kinda orbiting the asteroid but your orbital velocity is negligible. There is an emergency supply of food, water and air on Vesta if you can only get there.

You have 4 hours of oxygen left (including your emergency reserve) Will you make it? Hint: What is the gravitational acceleration that Vesta exerts on you? Hint: How long will it take before you are drawn to the surface of that asteroid?

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Problem #1: Calculate the value for "g" on the surface of Mars (Hint: remember, g is acceleration, NOT force)

Problem #2: You may have heard of the term "Gestation Orbit". A satellite is said to be in gestation orbit when it remains in orbit over the EXACT same location on Earth.

Talk with your group to suggest at least 2 different uses for such a satellite

Now calculate the height a satellite would have to be in order to remain in gestation orbit.

Compare gestation orbit with the altitude of the ISS

Next clockwise person leads the conversation now:

Problem #3: The Juno space probe is in a (not very circular) orbit around the great planet Jupiter. If we ignore that and just say that the orbit is circular and the probe is moving at .1056 miles/sec, how far above Jupiter's cloud tops is the Juno probe? (note: in turns out this value is much too small. If you're interested, go ahead and google "orbital speed of the juno probe" and Google will confidently spit out this value. It just turns out to be inaccurate. Go Figger)

Next clockwise person leads:

Be careful with your data on this one...

Problem #4) The solar system (literally: the stellar system containing the star named 'sol') is about 26,000 light years from the center of the Milky Way galaxy.

How fast are we orbiting around the galactic center? (Hint: The source of the force....Hint Hint: Assume ALL the mass in our solar system is concentrated in the sun, in other words ignore all planets, comets etc... Also assume all the mass of the galaxy is clustered at the Galactic Center. Considering the size of the black hole that resides there that isn't' quite as big a leap as you might think)

(It turns out there's a recent discovery that makes this a tad sketchy, but we'll talk about that during our closing).

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HOMEWORK

If time permits. Our security expert sent this around this morning. It is a very powerful video clip, but also rather intense