The Top Choice

Al grabs me as I step into his coffee shop. “Sy, ya gotta stop Vinnie, he’s using up paper napkins again, and he’s making a mess!”

Sure enough, there’s Vinnie at his usual table by the door. He’s got a kid’s top, a big one, spinning on a little stand. He’s methodically dropping crumpled-up paper wads onto it and watching them fly off onto the floor. “Hey, Vinnie, what’s the project?”

“Hi, Sy. I’m trying to figure how come these paper balls are doing a circle but when they fly off they always go in a straight line, at least at first. They got going-around momentum, right, so how come they don’t make a spiral like stars in a galaxy?”

Astronomy professor Cathleen’s standing in the scone line. She never misses an opportunity to correct a misconception. “Galaxy stars don’t spray out of the center in a spiral, Vinnie. Like planets going around a star, stars generally follow elliptical orbits around the galactic center. A star that’s between spiral arms now could be buried in one ten million years from now. The spiral arms appear because of how the orbits work. One theory is that the innermost star orbits rotate their ellipse axes more quickly than the outer ones and the spirals form where the ellipses pile up. Other theories have to do with increased star formation or increased gravitational attraction within the pile-up regions. Probably all three contribute to the structures. Anyhow, spirals don’t form from the center outward.”

My cue for some physics. “What happens in a galaxy is controlled by gravity, Vinnie, and gravity doesn’t enter into what you’re doing. Except for all that paper falling onto Al’s floor. There’s no in-plane gravitational or electromagnetic attraction in play when your paper wads leave the toy. Newton would say there’s no force acting to make them follow anything other than straight lines once they break free.”

“What about momentum? They’ve got going-around momentum, right, shouldn’t that keep them moving spirally?”

I haul out Old Reliable for a diagram. “Thing is, your ‘going-around momentum,’ also known as ‘angular momentum,’ doesn’t exist. Calm down, Vinnie, I mean it’s a ‘fictitious force‘ that depends on how you look at it.”

“Is this gonna be frames again?”

“Yup. Frames are one of our most important analytical tools in Physics. Here’s your toy and just for grins I’ve got it going around counterclockwise. That little white circle is one of your paper wads. In the room’s frame that wad in its path is constantly converting linear momentum between the x-direction and the y-direction, right?”

“East-West to North-South and back, yeah, I get that.”

“Such a mess to calculate. Let’s make it easier. Switch to the perspective of a frame locked to the toy. In that frame the wad can move in two directions. It can fly away along the radial direction I’ve called r, or it can ride along sideways in the s-direction.”

“So why hasn’t it flown away?”

“Because you put some spit on it to make it stick — don’t deny it, I saw you. While it’s stuck, does it travel in the r direction?”

“Nope, only in the s direction. Which should make it spiral like I said.”

“I’m not done yet. One of Newton’s major innovations was the idea of infinitesimal changes, also known as little-bits. The s-direction is straight, not curved, but it shifts around little-bit by little-bit as the top rotates. Newton’s Laws say force is required to alter momentum. What force influences the wad’s s-momentum?”

“Umm … that line you’ve marked c.”

“Which is the your spit’s adhesive force between the paper and the top. The wad stays stuck until the spit dries out and no more adhesion so no more c-force. Then what happens?”

“It flies off.”

“In which direction?”

“Huh! In the r-direction.”

“And in a straight line, just like Newton said. What you called ‘going-around momentum’ becomes ‘radial momentum’ and there’s no spiraling, right?”

“I guess you’re right, but I miss spirals.”

Al comes over with a broom. “Now that’s settled, Vinnie, clean up!”

~~ Rich Olcott

• Thanks for the question, Jen Keeler. Stay tuned.

Gravity and other fictitious forces

In this post I wrote, “gravitational force is how we we perceive spatial curvature.”
Here’s another claim — “Gravity is like centrifugal force, because they’re both fictitious.”   Outrageous, right?  I mean, I can feel gravity pulling down on me now.  How can it be fictional?

“Fictitious,” not “fictional,” and there’s a difference.  “Fictional” doesn’t exist, but a fictitious force is one that, to put it non-technically, depends on how you look at it.

Newton started it, of course.  From our 21st Century perspective, it’s hard to recognize the ground-breaking impact of his equation F=a.  Actually, it’s less a discovery than a set of definitions.  Its only term that can be measured directly is a, the acceleration, which Newton defined as any change from rest or constant-speed straight-line motion.  For instance, car buffs know that if a vehicle covers a one-mile half-mile (see comments) track in 60 seconds from a standing start, then its final speed is 60 mph (“zero to sixty in sixty”).  Furthermore, we can calculate that it achieved a sustained acceleration of 1.47 ft/sec2.

Both F and m, force and mass, were essentially invented by Newton and they’re defined in terms of each other.  Short of counting atoms (which Newton didn’t know about), the only routes to measuring a mass boil down to

• compare it to another mass (for instance, in a two-pan balance), or
• quantify how its motion is influenced by a known amount of force.

Conversely, we evaluate a force by comparing it to a known force or by measuring its effect on a known mass.

Once the F=a. equation was on the table, whenever a physicist noticed an acceleration they were duty-bound to look for the corresponding force.  An arrow leaps from the bow?  Force stored as tension in the bowstring.  A lodestone deflects a compass needle?  Magnetic force.  Objects accelerate as they fall?  Newton identified that force, called it “gravity,” and showed how to calculate it and how to apply it to planets as well as apples.  It was Newton who pointed out that weight is a measure of gravity’s force on a given mass.

Incidentally, to this day the least accurately known physical constant is Newton’s G, the Universal Gravitational Constant in his equation F=G·m1·m2/r2.  We can “weigh” planets with respect to each other and to the Sun, but without an independently-determined accurate mass for some body in the Solar System we can only estimate G.  We’ll have a better value when we can see how much rocket fuel it takes to push an asteroid around.

But there are other accelerations that aren’t so easily accounted for.  Ever ride in a car going around a curve and find yourself almost flung out of your seat?  This little guy wasn’t wearing his seat belt and look what happened.  The car accelerated because changing direction is an acceleration due to a lateral force.  But the guy followed Newton’s First Law and just kept going in a straight line.  Did he accelerate?

This is one of those “depends on how you look at it” cases.  From a frame of reference locked to the car (arrows), he was accelerated outwards by a centrifugal force that wasn’t countered by centripetal force from his seat belt.  However, from an earthbound frame of reference he flew in a straight line and experienced no force at all.

Suppose you’re investigating an object’s motion that appears to arise from a new force you’d like to dub “heterofugal.”  If you can find a different frame of reference (one not attached to the object) or otherwise explain the motion without invoking the “new force,” then heterofugalism is a fictitious force.

Centrifugal and centripetal forces are fictitious.  The  “force” “accelerating” one plane towards another as they both fly to the North Pole in this tale is actually geometrical and thus also fictitious   So is gravity.

In this post you’ll find a demonstration of gravity’s effect on the space around it.  Just as a sphere’s meridians give the effect of a fictitious lateral force as they draw together near its poles, the compressive curvature of space near a mass gives the effect of a force drawing other masses inward.

~~ Rich Olcott