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7. Uniform Circular Motion: Round and Round I Go

  1. General Observations

    What is uniform circular motion? Is it a police officer directing traffic? Sadly, (though that may be more interesting) the answer is no. I guess a police officer directing traffic would actually be uniformed circular motion, but once again, that's not what we want to discuss here.

    Uniform circular motion is actually a specific type of motion. An object is undergoing uniform circular motion if it is traveling at a constant speed while moving in a circle. For example, if you drive a car around a circle while maintaining a constant speed, you are performing uniform circular motion. If it makes you feel better, you could even wear a uniform while driving. The main requirements for uniform circular motion are a constant speed and motion in a circle. The object has to move in a circle and not some other shape. If the object is moving at a constant speed but is not moving in a circle, then it is not uniform circular motion.

    As you can tell in the above figure, the blue dot is moving in a circle. If you like, you can pretend that this is a top-down view of a car turning in a circle, where the blue dot represents the car. The radius is defined as the distance from the center (the pink dot) of the circle the blue dot is traveling around to the edge of the circle where the blue dot is located. The radius does not change as the blue dot moves around the circle. In addition, if the blue dot is traveling at a constant speed, then we can say the blue dot is undergoing uniform circular motion.

    If you were observant, you will notice that I said the object must be traveling at a constant speed while moving in a circle for the motion to be considered uniform circular motion. Why did I say constant speed instead of constant velocity? The reason lies in the definition of velocity. As you will recall, velocity is just speed with the exception that velocity also includes one more bit of information. Velocity also includes a direction in addition to the speed. This makes all the difference in the world. While an object undergoing uniform circular motion is moving at a constant speed, it is not moving at a constant velocity. This is because the direction of its motion continues to change as it moves around the circle. And, because the direction is always changing, its velocity is not constant. Remember, a constant velocity only occurs if both the object's speed and its direction are constant. The only way this can occur is if the object is moving in a straight line and at a constant speed.

    In the picture below, the blue dot is moving at a constant speed around a circle. In other words, the blue dot is undergoing uniform circular motion. Notice that while the speed is constant, the direction continues to change as the blue dot moves around the circle. In the figure below, the direction of the velocity (at any given point on the circle) is denoted by the red arrow.

    So, why did we spend an entire paragraph making the distinction that, while an object undergoing uniform circular motion is moving at a constant speed, it is not moving at a constant velocity? Well, think back to when we discussed Newton's 1st Law of Motion. In other words, if an object's velocity is not constant, what does that imply about the net force acting on that object?

    Newton's 1st Law of Motion tells us that, whenever an object's velocity is changing, there must be some external net force acting on the object to cause this change in velocity. Another way to think about this is to think of acceleration. Since the velocity of an object undergoing uniform circular motion is changing, it must be accelerating. This is because acceleration is defined as the change in velocity over time. Therefore, whenever an object's velocity is changing, it is experiencing an acceleration. According to Newton's 2nd Law of Motion, this acceleration must be caused by some external net force, once again, leading to the conclusion that an object undergoing uniform circular motion must be experiencing an external net force.

    Putting all these points together, we find that an object undergoing uniform circular motion has a velocity that is not constant. The velocity is not constant (even though the speed is constant) because the direction of the velocity continues to change as the object moves around the circle. Because the object's velocity is changing, we know the object is experiencing an acceleration as it moves around the circle. An object can only experience an acceleration when there is an external net force acting on the object because it is the external net force which causes an object to accelerate. In other words, when we see an object undergoing uniform circular motion, we know there must be some force acting on that object which is responsible for making the object move around a circle at a constant speed.

    Now that we know there is a force acting on an object undergoing uniform circular motion, it would be nice to figure out the direction of the force. Remember, a force has both a magnitude as well as a direction. We know there is a force acting on the object which is responsible for making the object move around a circle at a constant speed. However, in what direction must this force be applied?

    Well, once again, pretend you are at a park on a nice sunny day. It is very important to make the day sunny. While you are pretending, you might as well make it a nice sunny day, unless, of course, you are a vampire. Next, imagine that you have a ball attached to a string. Inexplicably, you decide to swing the ball around overhead. You do an extremely good job and keep the ball moving at a constant speed because you want to study uniform circular motion. In which direction do you have to pull on the string in order to keep the ball moving at a constant speed in the circle? As you might recall, we have already discussed this in a previous section, however, it doesn't hurt to review. If you have a ball attached to a string in real life, go ahead and try this little exercise right now. Remember to pay careful attention to the direction in which you have to pull the string in order to make the ball undergo uniform circular motion. If you don't have access to a ball, you can pretend that you are the pink dot in the center of the animated picture above and that the blue dot is the ball you are swinging around. Remember, this picture is an overhead picture. Using that animated picture, try to guess which direction you have to pull on the string to keep the ball moving around a circle at a constant speed.

    After a couple of spins, you should see that, in order to keep the ball moving at a constant speed around a circle, you need to pull inward on the string. Specifically, you need to pull the string toward you. In other words, you need to pull inward on the string along the radius (the line connecting the ball to the center of the circle it is moving around). In addition, you will notice that, if you don't change the radius of the circle, the harder you pull on the string, the faster the ball swings around. By using the same length of string, you can change the speed of the ball by adjusting how hard you pull on the string. The harder you pull, the faster the ball goes around. If you don't have a ball handy, you will have to trust me on this last issue. However, it should seem slightly possible that the speed of the ball can be increased by pulling on the string harder because, in a sense, you are making the ball turn faster around the circle by pulling on the string harder. We will have more to say about this last issue next time.

    (As an aside, in the situation where you are swinging a ball round and round overhead, you are not technically only pulling the string inward. You also need to pull the string slightly upward. This is because, in addition to making the ball go round and round, you also need to offset the force of gravity which is pulling the ball downward. That is why, in reality, you need to pull up and inward on the ball. Please refer to the picture on the left below. The picture on the left represents what we would have to do in reality to make the ball perform uniform circular motion. In addition to pulling inward on the string, we would have to pull slightly upward to offset the effect of the gravity pulling downward on the ball. For our discussion here, we will ignore the effect of gravity because we just want to concentrate on making the ball spin round and round. If we ignore the force of gravity pulling the ball downward, the only thing we have to do is pull inward on the string to make the ball perform uniform circular motion. For our discussion in this section, we will ignore the effect of gravity pulling downward on the ball. As a result, our situation will look like the picture on the right.)

      Thought Question:
    • In the situation above, where you are swinging a ball attached to a string overhead, what is the force responsible for making the ball undergo uniform circular motion? In other words, what is responsible for making the ball go round and round at a constant speed? (Once again, ignore the effect of gravity pulling the ball downward.)

      Well, this isn't really a very hard thought question. We've already discussed the answer above. The answer is that the force responsible for making the ball go round and round is you pulling on the string, which in turn is pulling on the ball. No surprises here.

      So, why did I make a thought question out of this? Well, it was mainly to emphasize that only an external force can make the ball undergo uniform circular motion. The ball cannot make itself do uniform circular motion. In addition, I wanted to emphasize that the force responsible for making the ball undergo uniform circular motion is not some new mysterious force. Any force which pulls the ball inward toward the center of the circle it is moving around can make the ball perform uniform circular motion. In this case, it just happens to be you pulling on a string attached to the ball.

    In the picture below, I have drawn a ball undergoing uniform circular motion. The pink arrow represents the direction of the velocity. The blue arrow represents the direction in which a force has to be exerted in order to make the ball undergo uniform circular motion. As you can tell, the force is directed inward toward the center of the circle. The force is along the line connecting the ball to the center of the circle. In other words, the force is directed inward toward the center of the circle, along the radius. It is very important to keep in mind that the ball cannot perform uniform circular motion by itself. There must be an external force pulling inward on the ball which causes the ball to go around in the circle at a constant speed. In the absence of a net force acting on the ball, the ball would move at a constant velocity. In other words, the ball would move in a straight line at a constant speed. Remember, this is what Newton's 1st Law of Motion tells us. It is only because there is a net force pulling inward on the ball that the ball moves around in a circle at a constant speed. Without this force (pulling on the ball), the ball would not be undergoing uniform circular motion. I cannot stress this point enough, which is why I've probably repeated it many many times already.

    Well, we've covered a lot of information, so let's summarize the main points so far as a quick review.

      Summary
    1. An object is undergoing uniform circular motion if it moves around a circle at a constant speed. This is just the definition of uniform circular motion.

    2. While the speed is constant, the object's velocity is not constant. This is because the direction of the velocity constantly changes as the object moves around the circle. (Please refer to the animated picture above.)

    3. Because the velocity is not constant, we know the object must be accelerating. This is true by definition because acceleration is defined as the change in velocity over the change in time. Therefore, if the object's velocity is changing, the object must be accelerating.

    4. Because the object is accelerating, there must be an external net force causing the object to accelerate. It is this external net force which is responsible for making the object move around a circle at a constant speed. Remember, without an external force pulling on the object, the object would not be undergoing uniform circular motion because Newton's 1st Law of Motion tells us that an object will maintain a constant velocity (i.e., moving in a straight line at a constant speed) unless an external net force is acting on the object.

    5. The direction of the force causing the object to undergo the uniform circular motion is directed inward along the line connecting the object and the center of the circle it is moving around. In other words, the direction of the force is toward the center of the circle, along the radius. Wherever the object is on the circle, the force required to keep it moving in a circle at a constant speed has to be directed from the object to the center of the circle it is traveling around.

    6. Finally, the force responsible for making the object undergo uniform circular motion is not some new mysterious force. There does not exist a mysterious force whose sole responsibility in life is to make things go round and round. I've listed some examples of forces which can make objects undergo uniform circular motion below. In the examples below, you should notice that the force responsible for uniform circular motion is not some new special force.

      For instance, if you are told that an object is undergoing uniform circular motion, that piece of information tells you nothing about the nature of the force or its origins. The only information you can get from this fact is that the force must be pulling inward on the object toward the center of the circle it is moving around, along the line connecting the object to the center of the circle.

        Examples:
      • In the case of a ball attached to a string, the force responsible for making the ball undergo uniform circular motion is you pulling on the string which then pulls on the ball it is attached to.

      • In the case of the earth orbiting the sun, the force responsible for keeping the earth in orbit is the gravitational force of the sun pulling on the earth.

      In both examples above, you should notice that the force responsible for making the objects perform uniform circular motion is not a new force. We have encountered these forces before in previous sections. For instance, we have already encountered gravity. We know that gravity is the force of attraction between any two objects that have mass. In addition, we know that gravity's sole purpose is not to make things go round and round. In this case, because the gravitational force is directed inward toward the sun while the earth is orbiting the sun, the force of gravity is responsible for making the earth perform uniform circular motion around the sun. Please refer to this section if you don't remember this example.

  2. Direction of Velocity

    At this point, we already know quite a lot about uniform circular motion. We know that a force is needed to make an object perform uniform circular motion. In addition, we know the direction in which the force must be applied on the object. Specifically, the force must be directed inward from the object toward the center of the circle the object is moving around.

    The one thing we left out of the previous discussion is the direction of the object's velocity as it undergoes uniform circular motion. From the picture above, it looks like the velocity is tangent to the circle around which the object moves as it undergoes uniform circular motion. For those unfamiliar with the term tangent, it just means that the velocity of the object is perpendicular to the line connecting the object to the center of the circle. In other words, the direction of the velocity is at a 90 degree angle with respect to the line connecting the object to the center of the circle.

    Well, as it turns out, the picture above is correct. As an object undergoes uniform circular motion, the direction of its velocity is always perpendicular to the line connecting the object to the center of the circle, regardless of where the object is on the circle. In other words, at any given point on the circle, the object's velocity is tangent to the circle.

    Now, this might seem a bit odd. Is there a way we can do a simple experiment to convince ourselves that the velocity of an object undergoing uniform circular motion is always tangent to the circle?

    Well, we can use our little experiment of swinging a ball around with a slight modification. Once again, we'll ignore the effect of gravity pulling the ball downward because we just want to concentrate on studying uniform circular motion. Using the picture above, pretend that you are the little pink dot in the center of the circle and the blue dot is the ball you are swinging around over your head. Remember, this is a top-down picture taken from overhead.

    Next, let's see what Newton's 1st Law of Motion has to say about our situation here. Newton's 1st Law tells us that an object will continue to move at a constant velocity unless a nonzero external net force acts on the object. Well, that doesn't appear to be very helpful on the surface. Let's reverse the situation around and consider an object that is not moving with a constant velocity (like an object undergoing uniform circular motion). If an object's velocity is changing, we know it is accelerating. As a result, we know there is a nonzero external net force acting on the object which is responsible for making its velocity change. So far, we haven't said anything new. After some time, let's say the external net force is somehow shut off. What can we say about the velocity of the object after the external net force is shut off?

    Well, this is precisely the concern of Newton's 1st Law of Motion. As soon as we shut off the external net force, the object's velocity no longer changes in response to the force. In fact, the object's velocity becomes constant, meaning that it will continue to move at a constant speed and in a straight line. Once again, this is nothing new. Next, consider the following question. How does the velocity of the object (after the external net force is shut off) compare to its velocity at the precise moment the external net force is shut off? To answer this question, it might be helpful to think of a specific case.

    • Example:
      Let's say you have a car that is undergoing a constant acceleration to the right of 5 (m/sec)/sec. Furthermore, assume you start out initially at rest. Because the car is accelerating, we know there is an external net force acting on the car. After 2 seconds, let's shut off the external net force acting on the car. What is the velocity of the car after 2 seconds?

      Well, from 0 seconds to 2 seconds, the car is accelerating. In fact, it reaches a velocity of 10 m/sec (to the right) after 2 seconds of time have passed. So, what is the velocity of the car after this time?

      To answer that, we need to think of Newton's 1st Law of Motion. After 2 seconds of time have passed, the external net force acting on the car is zero because we shut it off. As a result, the acceleration also becomes zero, after 2 seconds have passed, because it was the force which was causing the car to accelerate. Because of this, the velocity of the car will remain constant. As soon as the external net force becomes zero, the car's velocity becomes constant. This is what Newton's 1st Law of Motion tells us. In this example, the car's velocity is 10 m/sec (to the right) at the precise moment the external net force becomes zero. Therefore, the car's velocity remains at 10 m/sec (to the right) after the external net force is shut off.

      To repeat, the car speeds up from 0 m/sec to 10 m/sec (to the right) in 2 seconds of time. This is because there was an external net force acting on the car causing it to accelerate at 5 (m/sec)/sec to the right. After 2 seconds have passed, the external net force is shut off. At this point, the car no longer experiences an acceleration because the net force is zero. In other words, the car's velocity will remain constant after this time because the car no longer experiences an acceleration. Since the car's velocity was 10 m/sec (to the right) at the moment the external net force was shut off, the car maintains this velocity after the external net force is shut off. In other words, for any time after 2 seconds, the car's velocity is a constant 10 m/sec (to the right). It maintains this constant velocity because after 2 seconds, the external net force is shut off.

      This concept might seem a bit confusing so take your time with it. If you are having trouble with this concept, reread the above example and draw some pictures. In addition, go back and review Newton's Laws of Motion. Make sure you understand what went on in this example and convince yourself that it is true before reading further.

    So, the conclusion we get from the previous example is as follows. After the external net force is shut off, the object maintains a constant velocity. In addition, the constant velocity the object maintains (after the net force is shut off) is precisely equal to its velocity at the moment the external net force was turned off. What does this imply about the speed and direction of the object? Well, because the object's velocity (which includes speed and direction) after the net force is shut off is equal to its velocity at the exact moment the net force is shut off, the object's speed after the net force is shut off is equal to its speed at the moment the net force is shut off. In addition, the direction of the object's velocity after the net force is shut off is equal to its direction at the moment the net force is shut off. For example, let's say an object is accelerating. In other words, its velocity is changing. At some time, we decide to turn off the external net force which was causing the object to accelerate. At the moment the external net force was shut off, the object was moving to the right at a speed of 10 m/sec. As a result, the object will maintain this velocity of 10 m/sec (to the right) after the external net force is shut off. If you will notice, both the speed and direction after the net force is shut off is equal to the speed and direction at the moment the net force is shut off.

    You're probably shaking your head right about now. Perhaps, you're even thinking that this is one of those weird detours again. You might have even thought that this might have been more relevant in the section on Newton's Laws of Motion. Well, you are absolutely right on all accounts. The section on Newton's Laws of Motion would have been a perfect place for this discussion. That was a bit of short-sightedness on my part. However, the conclusion we just arrived at can also be used here to help us figure out the direction of the object's velocity while it is undergoing uniform circular motion. Can you think of a way that the above conclusion (in the previous paragraph) can be used to figure out the direction of the velocity of a ball we are swinging around? Think about that for awhile before reading the next paragraph.

    Okay, well, let's go back to considering the case of an object undergoing uniform circular motion. In particular, think of the case where you are swinging a ball around overhead at a constant speed. (Once again, we are going to ignore gravity for now.) As you swing the ball around, you are pulling inward on the string attached to the ball. The force you apply is the force responsible for making the ball move around the circle at a constant speed. As long as you pull on the string, the ball will continue to perform uniform circular motion. Next, at some point in time, let's say the string attached to the ball gets cut. When the string gets cut, what happens to the force pulling on the ball? Well, the force becomes zero because the string is no longer attached to the ball. After the string is cut, what does Newton's 1st Law of Motion have to say about the motion of the ball? As we have seen time and time again, when there is no net force acting on the object, the object's velocity remains constant. So, after the string gets cut, the ball's velocity remains constant.

    Well, so far, we haven't learned anything new yet. Or, have we? Think about the conclusion we arrived at above. The conclusion above states that, when the net force gets turned off, the object maintains a constant velocity (both speed and direction) which is equal to the velocity it had at the precise moment the net force gets turned off. Let's apply this conclusion to our particular case of uniform circular motion. At the moment the string gets cut, the force pulling on the ball becomes zero. Once the string gets cut, the ball will move at a constant velocity. It will no longer be performing uniform circular motion and will move in a straight line at a constant speed. More importantly, the ball's constant velocity (both speed and direction) will be equal to its velocity at the precise moment the string was cut.

    This last piece of information is exactly what we need to determine the direction of the ball's velocity as we swing it around. All we have to do is swing the ball around. When we swing the ball to the place where we want to determine the direction of its velocity, all we have to do is cut the string and observe the direction of the ball's velocity after it is no longer attached to the string. After the string is cut, the ball's velocity will be constant, but, more importantly, the ball's velocity (both speed and direction) will be equal to the velocity at the time the string was cut. By observing the direction of its velocity after the string has been cut, we can determine the direction of its velocity at the moment the string was cut because they should be the same. In other words, we can determine the direction of its velocity while it was swinging around a circle (at a particular place) by observing the direction of its velocity after the string is cut. We can do this because we know the ball's velocity (after the string is cut) is equal to its velocity at the moment the string was cut (while it was still undergoing uniform circular motion).

    By doing this several times and at different places around the circle, what we discover is that the direction of the ball's velocity as it undergoes uniform circular motion is, in fact, tangent to the circle it is moving around. In other words, the direction of the ball's velocity (at any given point on the circle it is moving around) is perpendicular to the line connecting the object to the center of the circle.

    Now, if you want to try this for yourself, instead of cutting the string, you can just let go of the string to shut off the net force pulling on the ball. It's a lot easier than cutting the string while you are swinging it around, and you get the same effect. If you are doing this on your own, the ball won't really move at a constant velocity after you release the string. In fact, it will arc downward toward the ground because there is the force of gravity pulling the ball downward. Unlike our fictitious example above, the ball will still have a force acting on it after you release the string because there is gravity. Remember, we had ignored gravity in our fictitious example above because we wanted to concentrate on just studying uniform circular motion without any added distractions. However, even though the ball does arc toward the ground after you release the string, you should still be able to tell that the ball's velocity (just after the string was released) is tangent to the circle it was moving around. As a result, we know the velocity of the ball is tangent to the circle it is moving around while it is undergoing uniform circular motion.

    In the picture below, the ball is undergoing uniform circular motion. After it completes one revolution, the string will be cut. At this point, the net force pulling on the ball is zero. Because the net force is zero after the string is cut, the ball's velocity remains constant after the string is cut. Because its velocity is constant after the string is cut, it will move at a constant speed and in a straight line. More importantly, the ball's constant velocity (after the string is cut) is equal to its velocity at the precise moment the string was cut (while the ball was still undergoing uniform circular motion). Using this fact, we see that the direction of the ball's velocity, while performing uniform circular motion, is perpendicular to the line connecting it to the center of the circle because the direction of its velocity, after the string is cut, is perpendicular to that line. Once again, this picture is taken from a top-down perspective, where you are the pink dot in the center swinging the ball around.

    Let's do a quick summary of the main points in this section.

      Summary
    1. Consider an object performing uniform circular motion. At any given point on the circle, the direction of the object's velocity is perpendicular to the line connecting it to the center of the circle it is moving around. In other words, the object's velocity is tangent to the circle.

    2. While the object is performing uniform circular motion, we can determine that the direction of its velocity is tangent to the circle it is moving around by using what we know about Newton's 1st Law of Motion. The procedure we used is as follows.

        Procedure Used to Determine the Direction of an Object's Velocity (while performing uniform circular motion)
      • Using Newton's 1st Law of Motion, we reasoned that an object's constant velocity (after the net force is shut off) is equal to its velocity at the precise moment the net force is shut off.
      • We then applied this conclusion to the case of uniform circular motion. If the force making the object perform uniform circular motion is shut off, the object's velocity will then become constant. More importantly, after the force is shut off, the object's velocity will be equal to its velocity at the exact moment the force was shut off.
      • At the point where the force is shut off, we can then determine the direction of object's velocity (while it is performing uniform circular motion) because it is equal to the direction of the object's velocity after the force has been shut off.

    This section may have been a little difficult. If it gave you some trouble, take a break and come back to it later. In addition, make sure you really understand Newton's 1st Law of Motion and its implications before rereading this section. If you decide to reread this section, take it slow and give it some time for the concepts sink in. In this section, I used Newton's 1st Law of Motion to suggest a way for us to show that an object's velocity (while it is performing uniform circular motion) is tangent to the circle it is moving around. If this argument was a bit unclear, the main thing to remember is the conclusion that the velocity is tangent to the circle.

Finally, here are the main concepts you should have gotten from this page. You should understand what uniform circular motion is. An object is undergoing uniform circular motion if it moves around a circle at a constant speed. While the speed is constant, the velocity is not constant because the direction of the object's velocity (as it moves around the circle) is always changing. Because the velocity is changing, we know the object must be accelerating. And, because it is accelerating, we know there must be some external force which is responsible for the change in velocity. As we saw in this section, this force has to be directed inward toward the center of the circle the object is moving around, along the line connecting the object to the center of the circle. The force making the object undergo uniform circular motion is not some mysterious new force whose sole purpose is to make things go round and round. For example, if you are swinging a ball around, the force of you pulling on the string is the force responsible for making the ball perform uniform circular motion. Lastly, if an object is undergoing uniform circular motion, the direction of its velocity is tangent to the circle around which the object is moving. These are the main points you should take away from this section.

Well, that's it for this section. As you may have noticed, we didn't really introduce any new formulae. In fact, all we really did was discuss some general observations about uniform circular motion. If you understood all of this, then you have a good understanding of uniform circular motion which you will need for the next section. Next time, we will introduce a formula which will allow us to calculate exactly how much force is necessary to keep an object moving under uniform circular motion. You already have a good general understanding of the physics of uniform circular motion. Next time, we will just be a little more specific.

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