Seesaw to The Stars

I look around the playground. “Where’s the seesaw, Teena?”

“They took it away. That’s good ’cause I hated that thing!”

“Why’s that, Sweetie?”

“I never could play right on it. Almost never. Sometimes there’d be a kid my size on the other end and that worked OK, but a lot of times a big kid got on the other end and bounced me up in the air. The first time I even fell off and they laughed.”

“Well, I can understand that. I’m sure you’ve been nicer than that to the littler kids.”

“Uh-huh, except for Bratty Brian, but he liked it when I bounced him. He called it ‘going to the Moon’.”

“I can understand that, too. If things go just right you come off your seat and float like an astronaut for a moment. I bet he held onto the handles tight.”

“Yeah, I just wasn’t ready for it the first time.”

“Y’know, there’s another way that Brian’s bounces were like a rocket trip to somewhere. They went through the same phases of acceleration and deceleration.”

“Uncle Sy, you know you’re not allowed to use words like that around me without ‘splaining them.”

“Mmm, they both have to do with changing speed. Suppose you’re standing still. Your speed is zero, right? When you start moving your speed isn’t zero any more and we say you’ve accelerated. When you slow down again we say you’re decelerating. Make sense?”

“So when Bratty Brian gets on the low end of the seesaw he’s zero. When I squinch down at my end he accelerates –“

“Right, that’s like the boost phase of a rocket trip.”

“… And when he’s floating at the very top –“

“Like astronauts when they’re coasting, sort of but not really.”

“… And then they decelerate when they land. Bratty Brian did, too. I guess deceleration is like acceleration backwards. But why such fancy words?”

“No-one paid much attention to acceleration until Mr Newton did. He changed Physics forever when he said that all accelerations involve a force of some kind. That thought led him to the whole idea of gravity as a force. Ever since then, when physicists see something being accelerated they look for the force that caused it and then they look for what generated the force. That’s how we learned about electromagnetism and the forces that hold atoms together and even dark matter which is ultra-mysterious.”

“Ooo, I love mysteries! What did Mr Newton tell us about this one?”

“Nothing, directly, but his laws gave us a clue about what to look for. Tell me what forces were in play during Brian’s ‘moon flight’.”

“Let’s see. He accelerated up and then he accelerated down. I guess while he was on the seesaw seat at the beginning the up-acceleration came from an up-force from his end of the board. And the down-acceleration came from gravity’s force. But the gravity force is there all along, isn’t it?”

“Good point. What made the difference is that your initial force was greater than gravity’s so Brian went up. When your force stopped, gravity’s force was all that mattered so Brian came back down again.”

“So it’s like a tug-of-war, first I won then gravity won.”

“Exactly. Now how about the forces when you were on the merry-go-round?”

“OK. Gravity’s always there so it was pulling down on me. The merry-go-round was pushing up?”

“Absolutely. A lot of people think that’s weird, but whatever we stand on pushes up exactly as hard as gravity pulls us down. Otherwise we’d sink into the ground or fly off into space. What about other forces?”

“Oh, yeah, Mr Newton’s outward force pushed me off until … holding the handles made the inward force to keep me on!”

“Nice job! Now think about a galaxy, millions of stars orbiting around like on a merry-go-round. They feel an outward force like you did, and they feel an inward force from gravity so they all stay together instead of flying apart. But…”

“But?”

“Mr Newton’s rules tell us how much gravity the stars need to stay together. The astronomers tell us that there aren’t enough stars to make that much gravity. Dark matter supplies the extra.”

~~ Rich Olcott

What’s that funnel about, really?

If you’ve ever watched or read a space opera (oh yes, you have), you know about the gravity well that a spacecraft has to climb out of when leaving a planet.  Every time I see the Museum’s gravity well model (photo below), I’m reminded of all the answers the guy gave to, “Johnny, what can you make of this?

The model’s a great visitor-attracter with those “planets” whizzing around the “Sun,” but this one exhibit really represents several distinct concepts.   For some of them it’s not quite the right shape.DMNS gravity well

The simplest concept is geometrical.  “Down” is the direction you move when gravity’s pulling on you.

HS cone
Gravitational potential energy change
for small height differences

A gravity well model for that concept would be just a straight line between you and the neighborhood’s most intense gravity source.

You learned the second concept in high school physics class.  Any object has gravitational potential energy that measures the amount of energy it would give up on falling.  Your teacher probably showed you the equation GPE = m·g·h, where m is the mass of the object, h is its height above ground level, and g is a constant you may have determined in a lab experiment.

If the width of the gravity well model at a given height represents GPE at that level, the model is a simple straight-sided cone.

Newton energy cone
Gravitational potential energy change
for large height differences
The h indicates
an approximately linear range
where the HS equation could apply.

But of course it’s not that simple.  Newton’s Law of Gravity says that the potential energy at any height r away from the planet’s center is proportional to 1/r.

Hmm… that looks different from the “proportional to h” equation.  Which is right?

Both equations are valid, but over different distance scales.  The HS teachers didn’t quite lie to you, but they didn’t give you the complete picture either.  Your classroom was about 4000 miles (21,120,000 feet) from Earth’s center, whereas the usual experiments involve height differences of at most a dozen feet.  Even the 20-foot drop from a second-story window is less than a millionth of the way down to Earth’s center.

Check my numbers:

Height h 1/(r+h)
× 108
Difference in 1/(r+h)
× 1014
0 4.734,848,484 0
20 4.734,844,001 4.48
40 4.734,839,517 8.97
60 4.734,835,033 13.45
80 4.734,830,549 17.93
100 4.734,826,066 22.42

rh lineSure enough, that’s a straight line (see the chart).  Reminds me of how Newton’s Law of Gravity is valid except at very short distances.  The HS Law of Gravity works fine for small spans but when the distances get big we have to use Newton’s equation.

We’re not done yet. That curvy funnel-shaped gravity well model could represent the force of gravity rather than its potential energy.  Newton told us that the force goes as 1/r2 so it decreases much more rapidly than the potential energy does as you get further away.  The gravity force well has a correspondingly sharper curve to it than the gravity energy well.

Newton force cone
The force of gravity
or an embedding diagram

The funnel model could also represent the total energy required to get a real spacecraft off the surface and up into space.  Depending on which sci-fi gimmickry is in play, the energy may come from a chemical or ion rocket, an electromagnetic railgun, or even a tractor beam from some mothership way up there.

No matter the technology, the theoretical energy requirement to get to a given height is the same.  In practice, however, each technology is optimal for some situations but forbiddingly inefficient in others.  Thus, each technology’s funnel  has its own shape and that shape will change depending on the setting.

In modern physics, the funnel model could also represent Einstein’s theory of how a mass “bends” the space around it.  (Take a look at this post, which is about how mass curves space by changing the local distance scale.)  Cosmologists describe the resulting “shapes” with embedding diagrams that are essentially 2D pictures of 3D (or 4D) contour plots.  The contours are closest together where space is most compressed, just as lines showing a steep hillside on a landscape contour map are close together.

The ED around a non-spinning object looks just like the force model picture above.  No surprise — gravitational force is how we we perceive spatial curvature.

~~ Rich Olcott