One way to use the Physics Footnotes repository is to insert footnotes individually into a lesson or presentation, to clarify, illustrate, elaborate, or to wake up your audience from time to time ðŸ˜‰ Another way to use the project is to search for all the footnotes on a particular topic and sew together your favorites to build a narrative that suits your personal style and curriculum needs. I’ll show you what I mean by taking Circular Motion as the theme. (Note that I’m not going to include here all the usual textbook definitions and derivations; the very idea of *footnotes* is to **supplement** the textbook.)

## Centripetal Force

It turns out that for a circle of radius \(r\) the *magnitude *of the centripetal force required to maintain uniform circular motion is given by \(F_c=m \frac {v^2} {r} = m\omega^2r\), where \(v\) and \(\omega\) are the linear and angular speeds (assumed constant) respectively. I mention both because, although they give the same answers, sometimes one is more mathematically convenient than the other.

### Footnote: Jahre Viking

The 458.45 m long Jahre Viking was the longest ship ever constructed. In good weather it could reach a speed of 30.6 km/h, and had a turning circle (i.e. radius) of 3 km. Calculate the g-force on crew members during such a turn. (Hint: They probably won’t need to strap themselves in!)

But what is the *direction* of centripetal force?

### Footnote: Direction of Centripetal Force

By thinking of a circle as a limit of a sequence of regular polygons, it’s not too hard to convince yourself that the force of the wall on the ball should point towards the center of the circle…

### Footnote: Lazy Physics Cat

Now because in uniform circular motion the centripetal force is perpendicular to the velocity of a body, it cannot do any work on that body (remembering that work is given by \(W=\vec F\cdot\Delta \vec X\)). So if the body experiences any resistance or friction on its way, the lost energy will need to be replenished by means of a separate tangential force. But don’t worry, if the engineering is good, it won’t take too much work…

## Sources of Centripetal Force

Now, all sorts of interactions can provide the necessary centripetal force to maintain circular motion. In the above case, the rim of the dish is continuously pushing the ball towards the center of the dish, while the cat periodically replenishes the energy lost due to friction. Let’s look at a few other interesting cases…

### Footnote: Extreme Skateboarding

Another very familiar example is the static friction between the road and your wheels as you execute a bend in the road. Supplemented, as necessary, by a touch of *kinetic *friction ðŸ˜‰

### Footnote: World’s Highest Swing

Another familiar source of centripetal force is tension. Like the tension in the ropes of the Nevis Swing in Queenstown, New Zealand; the highest swing in the world!

### Footnote: Coulombic Solar System

A not so familiar example is the electrostatic force keeping this charged water droplet in orbit around a knitting needle…

Before you try this one at home or in the lab I should warn you, though, that this had to be performed by an astronaut aboard a space-station to avoid gravity getting in the way!

### Footnote: Exploding Skateboard Wheel

But let’s not forget those situations where the interaction providing the centripetal force is unable to sustain circular motion if the speed gets too high. That’s what happened when these two guys decided to use a 60,000 PSI waterjet cutter to spin up a skateboard wheel to the tune of 45,000 RPM…

At such a speed, the glue used to hold the wheel together is unable to provide the centripetal force required to keep the material rotating at its original radius.

### Footnote: Leaning into a Bend

Physicists often like to simplify problems by treating a body as an infinitely small point mass. This is fine some of the time, but it’s going to get in the way of understanding the circular motion of real life objects like cars, bikes, and people. So let’s have a quick digression about this motorcyclist…

Now we all instinctively know to lean into a curve to avoid being thrown from our bike, but what is the right way to think about this necessity in terms of Newtonian mechanics? The answer lies in the *friction *between the road surface and the tyres, because it is this interaction that provides the centripetal force necessary to maintain circular motion. The trouble is that this frictional force also has a tendency to rotate the bike, as you see here…

When you lean into the curve, you shift your center of mass in such a way that your weight (which you can consider to act through your center of mass) provides a counter-torque to the friction which, if you lean over by the right amount, perfectly cancels the torque induced by friction…

## Circular Embankments

A vehicle can actually travel around a circular embankment without the help of friction, because gravity can indirectly help to provide the necessary centripetal force. Specifically, gravity pulls the car into the road, and the road pushes back with a normal reaction force. That normal reaction force has a component toward the center of the circle, and provided you travel at the right speed (given the angle of the embankment and its radius), it will be sufficient to maintain the circular motion without the help of friction.

If you’re designing such an embankment so that cars can travel safely around it at speed \(v\), it turns out the angle of the embankment needs to be given by \(\tan\theta = \frac {v^2} {gr}\). (Of course, in reality, the road will have friction and this will lead to a range of safe speeds.)

### Footnote: Velodrome Design

This is a snippet from TeamGB’s world record race at the 2012 London Olympic Games. The race is taking part inÂ London’s Velopark, a velodrome built especially for the event…

By keeping their bikes perpendicular to the track surface, the normal reaction (rather than friction) provides most of the centripetal force, thereby avoiding slippage. The embankment angle around the curved segment of this particular track, for example, is 42 degrees, which allows the cyclists to execute the turn at speeds in excess of 80 km/h. If the cyclists were to attempt this turn on a flat surface (made of exactly the same material), they would be unlikely to reach 50 km/h before slipping out of the circuit.

### Footnote: Worker in a Hole

Now suppose you are this guy, stuck at the bottom of a slippery sloping hole without a ladder. What to do? Well, the radius of the circle increases as you go up, and for a given radius \(r\) the normal force could support a speed \( v=\sqrt {gr\tan\theta}\), so all you have to do is accelerate as you ascend keeping at or above this threshold. If the hole is too deep or too steep, you lose…

### Footnote: Leonard’s Olive Trick

**Discussion Question**: Ok, so far so good. But how does Leonard from the Big Bang do his olive-in-glass-without-touching-it-to-impress-girl trick, when the walls of the glass slope in the opposite direction to the embankment?

The physics of the embankment problem is pretty much all we need to tackle another phenomenon, first discussed in detail by Newton in his Principia…

### Footnote: Rotating Bucket

If you take a bucket of liquid and rotate it steadily about its central axis, you’ve probably noticed that the surface of the water eventually stabilizes, forming a smooth concave surface. Here’s a particularly striking example…

It turns out that when you do the math, the shape of the surface of a container of fluid having an angular rotational speed of \( \omega \) is given by \(y= \frac {\omega^2} {2g}x^2 \), where the point \( (x,y) = (0,0) \) is taken to be the lowest point on the fluid’s surface.

And no it’s not just in ‘theory’. This simple but elegant experiment bore it out beautifully…

### Footnote: Liquid Mirror Telescopes

Because telescope mirrors need to have parabolic surfaces, they are sometimes made by rotating a liquid until its surface is almost perfectly smooth, and then cooling the liquid sufficiently for it to solidify in that shape.

As a matter of fact, sometimes astronomers skip the last step, and just use the rotating liquid (usually mercury) as a mirror…

## Centrifugal Force

When you are constrained to move in a circle at high speed, you feel yourself being pressed hard against the constraint, and we often attribute this to a radially outward force which has come to be called ‘centrifugal’ force. Now whilst your experience is certainly not imaginary, describing this sensation as a radially outward force does play havoc with Newtonian mechanics, and so we have to analyze the situation very carefully.

In particular, ask yourself the following question. If I were to remove the constraining (centripetal) force from an object moving in a circle, where does the outward (centrifugal) force go?

### Footnote: Spinning Sparkler

In this clip, a sparkler has been radially attached to a drill, by bending it at its attachment point. As the sparkles leave the constrained system, they fly off in straight lines, from which we infer the absence of force. Now we know where the centripetal force went; it disappeared because the object broke away from its constraint. But where did the centrifugal force go?

The answer is that a centrifugal force is not due to a real physical interaction between bodies (like that of a compressed spring, for example), but rather to a relative choice of perspective or ‘reference frame’. You *include* it if you want to consider a rotating frame to be a rest frame, and you *exclude* it if you choose an inertial frame as a rest frame. We say such a force is non-Newtonian, and we really need to be very careful if we do physics that way.

Having said that, provided you know what you’re doing, the use of centrifugal force is a very handy tool. In fact, if you treat gravity in the usual Newtonian textbook way, you use this trick all the time…

### Footnote: Gravity as Centrifugal Force

But before you say it is *wrong* to do physics that way, I’d like to point out something that a lot of people don’t realize. According to the General Theory of Relativity, gravity is *exactly* that type of force! In other words, gravity is the name we give to the force pressing us into the ground by virtue of the fact that a reference frame attached to the ground is actually non-inertial…

Despite what the Newtonian picture assures us over and over again, it is actually a freely falling observer who is inertial.

### Footnote: Pouring Water During Barrel Roll

In fact, it is often convenient when describing circular motion from the point of view of the moving object, to think of your motion resulting in the appearance of a gravitational field which points radially outwards. If Einstein is right, this is no more wrong than (or perhaps equally wrong as) the way we handle ordinary gravity all the time!

In this way, the guy in this jet can make perfect sense of his ability to pour water into a cup no matter what his orientation throughout a barrel roll…

### Footnote: Playing Catch on a Rotating Platform

Let’s take this little experiment, for example, in which affixing a camera to the rotating platform trains us to consider the platform frame to be at rest…

From the woman’s perspective, there is a gravitational field emanating outward from the center of the platform. She shouldn’t be surprised by the fact that she can throw the ball with one hand and catch it with the other, much like under ordinary gravity, except that the ‘field’ will be non-uniform making the trajectory non-parabolic.

### Footnote: Rotating Seedlings

The ability of a plant to direct its growth along gravitational field lines is calledÂ *gravitropismÂ *or, sometimes,Â *geotropism*.

Now if a plant grows on a rotating turntable, the result (if what I’ve been telling you is true) should be empirically equivalent to a plant growing with a gravitational field that is tilted from the vertical (the net field being the usual one added vectorally to the negative of the centripetal acceleration). Therefore we should see the plants growing inward as they grow vertically, with an angle determined by the rotational speed of the turntable.

But does this really happen? Hereâ€™s the short answer:

Anyway, you get the drift right? The way you would glue these footnotes together (or even the ones you would choose in the first place) would be very different to the narrative I’ve woven. And that of course is the whole point. A flexible system of footnotes you can select and organize in any way you like.

Well, that’s what I’ve been working on for *months* now at least. There’s a long way to go. If you’d like to support the project, you can have full access to the repository any time you like, starting from right now! **Click here for more**.

If that doesn’t interest you, feel free to follow along on Facebook, Twitter, or (most importantly) by email below, and add them to your own storehouse of goodies ðŸ˜‰

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