LEARNING OBJECTIVES:

I will be able to explain why centripetal force DOES exist and why centrifual force is a misnormer to an articulate 9th grader during today's class.

WORDS O' THE DAY:

• Centripetal ("towards the center")
• Centrifugal ("away from center")
• Gravitational Constant: 6.674 × 10^-11 Nm²/kg²

FORMULAE OBJECTUS:

• T² ∝ A³: The square of the period (in years) of an object orbiting the sun is proportional to the cube of the average distance to the sun (in years). This is kind of archaic in that we rarely see the proportional symbol anymore.

• T² = a³: The square of the period (in years) of an object orbiting the sun is approximately equal to the cube of the average distance to the sun (in years). We MUST keep in mind this is an observational relationship. Although it gets us close (in most instances) it is not an exact value so the "=" sign isn't really appropriate although it is widely used.

• M_sT² = a³: This version is still approximate but it allows us to substitute in the mass of *other* stars as long as we measure the mass of the other star in terms of the mass of the sun being 1.00

• T² = (4π²/GM)(a)³ = This version is much more accurate is often referred to as Newton's version of Kepler's Law. Notice ALL values must be in SIU

• v²/r: centripetal acceleration -- The acceleration of an object in circular motion is determined by the velocity (squared in m/s) divided by the distance to the center of that circular motion (in meters)
• mv²/r: centripetal force -- this is nothing more the F = ma for an object experiencing circular motion.
• F_g = Gm₁m₂/r²: This is Newton's famous equation for gravitational attraction. 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:

JUST FYI - I've combined a couple of lessons here so this will definitely continue on Monday!

WHEW!

Now let's have a couple of partnerships come up and present their work on the whiteboard.

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We're going to change direction now to discuss objects in circular motion AND objects that orbit the Earth!

Speaking of which, SpaceX just launched Starship 8. What happened?

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We need to start out by making sure we are comfy with centripetal force/acceleration as opposed to centrifugal force. The former exists, the latter does not. Let's discuss!

 

 

 

Keep in mind an object moving in a circle does NOT magically, suddenly develop a mysterious force that pushes outwards.

Mr Swanson was kind enough to give me a turntable a few years back, allow me to demo!

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We MUST keep in mind the SOURCE of the centripetal FORCE that KEEPS an object moving in a circle.

For example, what is the SOURCE of the FORCE that keeps a sattelite moving around the Earth?

What is the SOURCE of the FORCE that keeps an object tied to the end of a string from flying off into space? (I'll demo!)

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The centripetal force that an object in circular motion experiences is determined using Newton's 2nd Law:

F = ma

However, that is for an object moving in one direction. The calculation for an object moving in circular motion is

F = m(v²/r)

Where v is the circular velocity in meters/sec and r is the distance to the center of the object tugging on the "leash" of the object

 

To reiterate, centripetal force is a description of the force an object experiences in circular motion, it is NOT the source of that force. There is NO magical force that magically flashes into existence when an object starts to move in a circle.

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You swing a horseshoe with mass 2.50 kg around and around above your head on a rope of length 1.75 meters. At first it's a bit ragged and the horseshoe sort of bobs and weaves but you soon develop a more gentle rhythm of tugging on the rope (pulling the horseshoe towards your hand) and the horseshoe moves in smooth circles.

• Do a Newton's First analysis on the horseshoe (hint: gravity is a non-issue here. Why is that by the by?).
• Now add in a description of the motion of the horseshoe.
• If the horseshoe has a circular velocity of 12.0 m/s (in other words it travels an arc of 12.0 meters in a circular direction every second) how much force are you exerting on the rope?

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Human beings do badly in the absence of gravity. Bones get brittle, muscles get weak. In order to simulate gravity on deep space missions, engineers propose having part of a space ship rotate... like this:

Work with your group to solve this problem (move your mouse over the dark area between the sections to see my solution)

1) Estimate the radius of the rotating section of the ship shown in that clip

5.0 m or so

2) Use that value to determine the circumference of the rotating ring (hint: 2Πr!)

~31.4 m

3) Estimate/measure/observe the speed of rotation for that drum (this one is a bit tricky. It might help to measure the time for one full revolution of the compartment. That will give you seconds/meter... hmmmmmmmmmmmmm)

~25 sec per revolution

But each revolution is about 19m so that means the drum is rotating at about 31.4m/ 25 sec

=> 31.4m/25sec

=1.26 m/s

5) Calculate the amount of acceleration (v²/r) that man experiences

(1.26 m/s)²/(5.0 m)

WHOA.... did the filmmakers make a bit of a boo boo here?

Why might that have occurred?

6) Compare that value to 'g'

7) The filmmakers almost certainly knew that this scene didn't quite match the physics they were trying to portray. To find out why they made the choice to do that scene that way, let's calculate how large "r" would have to have been (rotating at the same speed) to give our astronauts an artificial gravity of 1.0 g

v²/r = A_c

r = v²/A_c

r = (1.26m/s)²/(9.81 m/s/s)

r = .162 m

Clearly the filmmakers weren't going to make a rotating drum a few centimeters in diameter. They wanted to illustrate just how one might go about modeling a true space flight but making any sort of scale model was just way, way, way too expensive. So, they opted for building a rotating drum they could make on set to *model* how it might work in outer space, knowing all the while the science wasn't accurate (and knowing also that almost no one in their audience would bother to check the math, or even know how to do that!)

8) A somewhat more realistic value for 'r' would be in the neighborhood of 100. meters (roughly the length of a football field). How fast would that drum have to turn so that resulting acceleration experienced by the astronaut would be a more reasonable .50 g?

v²/r = A_c

v² = rA_c

v= sqroot(rA_c)

v = sqroot(100.m)(.50)(9.81m/s/s)

v = ~22 m/s

 

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HOMEWORK