Kevin has edited encyclopedias, taught history, and has an MA in Islamic law/finance. He has since founded his own financial advice firm, Newton Analytical.
Sure, Newton's Second Law of Motion works well in one dimension, but what happens when you put it on a curve? In this lesson, we'll see how the Second Law applies with respect to Uniform Circular Motion.
Newton's Second Law Spun
By now, you're probably pretty familiar with Newton's Second Law of Motion, the one that says that force is equal to mass times acceleration. You've seen how a bullet accelerating at a very fast rate can do as much damage as a spear, with a much greater mass, that has a much lower acceleration. However, what's with that word 'acceleration?' Can't we just say 'velocity?' Unfortunately, it's not that easy. One of the main reasons that it isn't so simple is because once we apply Newton's Second Law to motion, velocity is constantly changing. In other words, for us to be able to explain what is going on, we have to use acceleration. In this lesson, we're going to do exactly that, explaining how Newton's Second Law of Motion explains what is called uniform circular motion. Uniform circular motion is the act of motion in a circle around a very precise point.
Usually, when we use Newton's Second Law, the a for acceleration is pretty easy to define. More often than not, it is in terms of meters per second squared in a given direction, or some other similar unit. However, because we're moving around a circle, we need a different way to find the acceleration. Otherwise, we'd be stuck defining a new direction every millisecond. To get over this, we state that acceleration is equal to the speed squared divided by the radius. This can also be found by squaring the angular speed and multiplying it by the radius, depending on what information you have available. We simply plug that new number into our usual equation, meaning that now F is equal to m, for mass, times s for speed, squared, divided by r, for radius.
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Some people may look at that math, shrug their shoulders and instantly internalize what it means. Unfortunately, I'm not such a person, so let me explain it in English. The tighter your curve, the greater your change in acceleration. This makes sense, after all. If you want to travel around a ten-meter diameter dome you can see that your direction, and therefore your acceleration, are both constantly changing. Meanwhile, you don't really feel your direction changing when you're walking on the surface of the Earth, because it's nowhere near as tight of a curve. Speaking of tight curves, have you ever felt as though you were being flung out of a curve and ended up landing on the side? Despite what people tell you, that's not so-called centrifugal motion. Instead, it's just inertia. At each instant, your body is pushing you out in that direction. Meanwhile, there is a force acting to pull you back in called centripetal motion. That is the equal and opposite action of all the other forces pushing you out. This is what keeps you from flying out of the circle. Let's pretend you are on a merry-go-round. You'd better hold on, because if you don't, you'd be flung off, right? Now, that sensation of being flung off, if you don't hold on, is not centrifugal motion, but is, instead, inertia. Inertia, as you'll remember, is the desire for an object to keep moving in the same direction that it is moving at that very instant. It is the reason that when you throw a ball, it goes straight rather than follow the circular pattern of your pitch. Meanwhile, there's also a force that keeps pulling you towards the center. Namely, it is centripetal motion. In our example of the merry-go-round, it is the force that you have to exert to hold on. When a merry-go-round is spinning slowly, you don't have to hold on as hard. This is because the centripetal force can be lower. Meanwhile, if it is spun faster, you have to hold on with a much greater force.
Let's look at a couple of examples to make sure that we've got this down. We'll do one with math, and one with just the concept. First, the conceptual example. Let's say that you are a race car driver going into a turn. What forces are acting on your car? First of all, you've got inertia taking your car in a straight direction from wherever you are at that particular moment. Second, you have centripetal force pulling you back. So, how do you think that centripetal force is able to counteract your speeding car? If you're lucky, the curves are banked. This means that less of the inertia force can pull on your car, because it is now going against the normal force exerted by a surface against the object on it. Now let's try some math. Let's say that you were looking at the force exerted on a 5 kilogram cat that some mischievous child has put on a merry-go-round. The merry-go-round has a radius of 1 meter, and the child is moving it around at 5 meters per second. Let's use our force formula of F=ms^2/r. Five squared is 25, divided by 1 is still 25. Multiply that by 5, and you'll find that 125 newtons of force are acting on our poor cat.
In this lesson, we look at Newton's Second Law of Motion and uniform circular motion. Remember that Newton's Second Law of Motion states that force is equal to mass times acceleration, and uniform circular motion refers to perfectly circular motion around a center point. We calculate this by substituting the square of speed divided by radius for the acceleration, although the general theme of the tighter the radius, the more force is exerted, holds true.
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