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Lets start out with a definition. Uniform circular motion is when you have something moving at a constant speed in circle. The important thing to remember is that the speed is constant the velocity is not. Why isn't the velocity constant? Well what is the difference between speed and velocity? That's right, speed is a scalar (without direction) quantity and velocity is a vector (with direction) quantity. While the speed may be constant, say one meter per second, the velocity is always changing. This is because in order to move in a circle, you have to always be changing direction. I have a funky little animated gif to help show this idea.
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Hypnotising isn't it? |
Allow me explain what you are looking at. The blue arrow is a velocity vector. It represents the velocity of an object moving around the circle. The object would be at the base of the arrow. No matter where we look at the velocity vector, it's length is the same. If you remember from the vectors lesson, the length of a vector is its magnitude, in this case that translates into the speed. Since the length of the vector (its magnitude) never changes, we can say that the speed of the object is always the same.
Direction is another matter. Obviously the direction is always changing. Because the direction changes, we have a different vector, so the velocity of the object is not always the same. This is important to remember. Teachers like to trip up students by asking question like: Is the velocity of an object in uniform circular motion uniform? Your first impulse is to say yes, but resist the temptation and remember. Speed is uniform, velocity is not. All right I'll stop saying that and lets move on.
The kind of velocity shown above is often called tangential velocity, because it is tangent to the circle. OK, so what does tangent mean? It is a math term that means the line only touches the circle at one point. OK enough of this crap! You want the formula that will get you answers, so here it is.
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This says that the speed of an object moving in uniform circular motion is equal to two times pi times the radius of the circle divided by the period. In uniform circular motion we replace time with period. The period is the time it takes for the object to go around the circle once.
If you take the time to think about it, it should make sense. Velocity is displacement (or distance in some cases) divided by time right? So if you look at the top of the equation, the 2 Pi r part, does that look familiar? Well if it doesn't, 2 Pi r will give you the circumference of a circle. That is the distance that our object is traveling, so that is the same as the 's' in the more familiar v=s/t. Now the 'T' is the period, which is just the time it takes to go around the circle, or the time it takes to cover the distance on the top of the equation. So upon closer inspection v=2 Pi r / T is really just the old familiar v=s/t in fancy clothing.
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Anytime there is a change in a velocity there must be an acceleration. Why? 'Cuz that's what accelerations do with their free time, they change velocities. When an object is moving in uniform circular motion, its velocity is always changing, so there must be an acceleration. It turns out that the acceleration is toward the center of the circle. It's kinda an uncomfortable thought because when we think of acceleration we usually associate it with a change in speed, but that isn't happening here. The acceleration is only changing the direction.
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Here is the only way I can understand this concept of acceleration toward the center. You may want to read the projectile motion lesson before reading this.
Imagine you are standing on a flat plane. If you fired a gun horizontally the bullet would go out the nozzle, travel forward and also fall (accelerate) toward the ground because of gravity. So far so good.
Now pretend that we are no longer on a flat plane. Pretend that the plane curves down at the end. If this is so, it means that the bullet has to fall farther at the end. This is because the ground is curving away from the bullet. If we are able to curve the ground into a circle, and give the bullet an exact speed, will be able to put the bullet in orbit. Why? OK here we go. The bullet is moving in two directions, forward and downward. As the bullet moves forward it also falls down. If the ground curves away from the bullet as it falls, the bullet has farther to fall, but by the time it falls that distance, it has also moved forward. If it has moved forward, the ground has curved away more, so it has farther to fall. Can you see where I'm going with this? So the bullet is always falling (accelerating) toward the ground, and because it is also moving forward, and the ground continuously curves away, it never hits the ground. I hoped it helped, but if it didn't you can always just memorize the formula. That's what most people do.
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The centripetal acceleration is equal to the square of the object's speed divided by the circle's radius.
Centripetal Force
I don't really have much to say about centripetal force. Like all forces, it is a vector that points in the same direction as the acceleration. In uniform circular motion the centripetal force is what is keeping the object in circular motion in the first place. If you have ever swung a weight attached to a string about your head, the centripetal force is the pulling force you exert on the weight. Here is the formula.
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The centripetal force is equal to the mass times the velocity squared divided by the radius of the circle.
Like the formula for velocity, Fc=mv2/r is really just F=ma all dressed up. F=ma tells us the force for a given mass and acceleration. If we want to find the centripetal force, all we do is substitute the regular acceleration, with the centripetal acceleration ac=v2/r. So F=ma becomes Fc=mv2/r.
If any of the above was unclear, or if you have any comments or suggestions, please E-mail me!
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