2.9 Circular Motion and Orbits

Critical Questions:

• What keeps the Earth orbiting the Sun instead of shooting off into space?
• How do figure skaters achieve such elegant spins?

As I mentioned earlier, we spend most of our time on a giant hunk of rock that is moving in a circle (approximately) around the sun, just like every other planet in the solar system. And the sun is circling the center of the galaxy. In fact, if you spend enough time looking out into space, you start to realize that almost everything out there is either spinning around, orbiting something else, or both. And understanding that motion represents an important step in the understanding of physics as a whole.

What makes an object move in a circle? Perhaps a more useful question to ask first is, why don’t objects usually move in circles? We know from Newton’s First and Second Laws that objects keep moving in a straight line (or stay stationary) unless an external force causes them to speed up, slow down, or change their direction of motion. And yes, if you’ve been reading carefully, you’ll notice that this answer amounts to a fancy way of saying, “Because.” A slightly better answer might be, “Because space seems to be aligned in straight lines rather than curved ones,” but the truth is, that answer is just yet another way of restating the question.

So matter moves in straight lines unless a force changes its motion. Now imagine a ball on the ground. How would you get it to move in a circle? The answer is that you would get it rolling forwards, and then constantly deflect its path so that it turns to one side. Eventually, if it keeps turning, it’ll come back to where it started. And if you looked at that circle from above, you’d see that your sideways push on the ball was always directed towards the center of the circle.

The name for a force that pulls an object in towards a center point is a ‘centripetal’ force. This is not to be confused with the ‘centrifugal’ force, which I’ll explain in just a minute. It is also not a new variety of force, like gravity or electromagnetism. Any force can be a centripetal force, as long as it pulls in the right direction. In the case of a ball on a string, the centripetal force is the tension force applied by the rope; for a car turning a corner, the centripetal force is friction between the road and the wheels.

We can stop here and mention that we’ve now solved the multiple choice problem from earlier on – the one about the ceiling fan and the golf ball. The ball first moves in a circle due to a centripetal force (which was provided by the tape and the fan blade). When that force disappears, the ball will immediately fly off in a straight line, no matter how long it had been experiencing circular motion before then. This is simply because there is no longer a force that is causing it to turn.

We’re complicating things a bit, once again, by thinking about things on Earth. Because of friction and air resistance, the objects we’re talking about are always going to tend to slow down, and so they also need a forwards push to keep them moving. That’s why you don’t turn the wheels of your car 90 degrees to the left when you turn a corner: you need the wheels to push you forwards and to the side. When you attach something to a string and whip it around your head, you mostly pull the object in towards the center of the circle (centripetal force), but you also move your hand just enough to be pulling it slightly forwards as well, to counteract air resistance. The name for this forwards force is the ‘tangential’ force, because it pulls at a tangent to the circle being described by the moving object[1. If you don’t know what a tangent line is, imagine holding a stick up to the edge of a hula hoop. The line the stick makes is ‘tangential’ to the circular hoop.].

Up in space, however, there is no friction or air resistance to slow things down. If a planet is moving forwards and has a centripetal force pulling it constantly to the side, it can keep going like that forever, always turning towards the star it’s orbiting but never coming any closer. This centripetal force is provided by gravity.

The nice thing about gravity is that it will always pull towards the center of the circle: the sun is at the center of the Earth’s orbit, for example, and the planet is attracted directly towards it. Gravity is, in fact, the centripetal force responsible for all of the orbits in space: the moon orbits the Earth, as do man-made satellites; all of the planets orbit the sun, and all other planets orbit other stars; the sun and all of the stars in the galaxy orbit the galaxy’s central point (which contains a very large amount of mass, and probably a black hole or two), etc.

So what about the centrifugal force? Imagine again that you’re inside a car as it’s making a sharp left turn. What do you feel? Carsick, maybe, but you’re probably also feeling pulled to the right. This is the centrifugal force at work. Certain kinds of people like to tell you quite smugly that this isn’t a force at all, and in some ways they’re right – factually, if not morally.[2. You can smugly reply by saying that the centrifugal force really is a force if you consider the whole problem from a rotating, non-inertial frame of reference, but pray that they don’t ask any follow-up questions. (See also this xkcd comic.)]

The only reason you feel pulled to the right is because your body obeys the law of inertia. You were travelling straight forwards, and you will tend to keep doing so. Meanwhile, the car surrounding your body is turning left, thanks to friction. You go straight, but the car goes left; from your perspective, it seems like you’re getting pulled to the right. The term “centrifugal force” is a misnomer because there is no force pulling you that way. However, this phenomenon has proven to be a useful one – it’s what gets water off the lettuce in your salad spinner, and it separates heavier substances from lighter ones in machines called centrifuges.

The last thing we’ll talk about is spinning. Most of the physics of spinning objects is pretty obvious, but there’s one intriguing thing that is not commonly understood. Imagine a figure skater balanced on the tip of a skate blade and spinning gracefully in place as an instrumental version of something by Celine Dion plays over the loudspeaker. The skater’s arms are extended out to the side, but as you watch, she brings them in closer to her body, and all of a sudden, she is spinning much faster than before.

To explain this phenomenon, I’ll ask you to imagine a spinning record. Draw two dots on the surface – one close to the center (dot A) and one out near the rim (dot B). Now watch the dots moving for a while. What you’ll see is that dot B moves much faster than dot A. This is because in the same amount of time, dot B has to go in a much bigger circle than dot A does.

Now we’ll further damage the record by taking something heavy, like a paperweight, and placing it right on top of dot B. We’ll get the record spinning around so that the paperweight is moving at the same speed as the dot was before. Now take the paperweight and push it in towards the middle while the record is still spinning. (I’ll admit that this would be hard to do smoothly in real life, but try to imagine it anyway.) The paperweight is now sitting on top of dot A, but because of the law of inertia, it is trying to move as fast as it was at dot B – that is, faster than the record is moving beneath it. So the rock pulls the record forwards a bit, which increases the record’s rate of rotation.

This is just like what the figure skater did when she pulled her arms in close. They were moving quickly because they were out at the edge of the circle of her spin, so when she pulled them in closer, they pulled the rest of her along until she was spinning faster than before.

There’s quite a bit more to be said about rotation and circular motion, of course, but this is a good place to stop for now, because the next chapter will go into all kinds of detail about gravity and the orbits it can cause.

Big Ideas:

• Circular motion requires a centripetal force, which constantly turns the object in towards the center of the circle.
• For orbits in outer space, the centripetal force is gravity.
• The centrifugal “force” is not really a force. Objects moving in a circle feel pulled to the outside of that circle because of the law of inertia, which says that moving objects tend to continue in straight lines.
• Spinning objects can change their rates of spin by moving parts of themselves closer to or farther from the center of their spin.

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Next: 3.1 – Introduction to Gravity and Orbits

Previous: 2.8 – Friction and Air Resistance