• Uniform Circular Motion: (Continued)
  • How Much Force Do I Really Need?
    • Relationship between Force and Speed
    • Relationship between Force and Radius
    • Summary

7. Uniform Circular Motion: (Continued)

1. How Much Force Do I Really Need?

   First of all, let me apologize for the less than imaginative title of this section, but this title does aptly describe what we will discuss in this section. Next, before proceeding, make sure that you have a good understanding of the physics of uniform circular motion. If you are still unclear about uniform circular motion, please reread the previous section and give it some careful thought before coming back here. We will use a lot of the previous information in this section.

   Without further ado, let me introduce the formula which tells us how much force is needed to make an object undergo uniform circular motion.

   
   where

   • F_c = the amount of force needed to make the object undergo uniform circular motion at a constant speed,v, and at a radius, r.
   • v = the speed of the object as it performs uniform circular motion
   • r = the radius of the circle around which the object is moving
   • m = the mass of the object undergoing uniform circular motion

   

   First of all, you will notice the force, , has a subscript attached to it. The subscript, c, tells us the force is always pointing inward from the object toward the center of the circle around which it is moving. The subscript, c, serves to remind us that the force, , is always pointing toward the center of the circle.

   Ok, back to the formula. Once again, this formula only applies to an object moving under uniform circular motion. If the object is not performing uniform circular motion, i.e., moving around a circle at a constant speed, then you will have to look elsewhere for help because this formula won't do.

   Next, this formula assumes you have an object moving around a circle of radius, r, while moving at a constant speed, v. In the picture above, you can tell the blue dot is moving around a circle of radius, r. The radius is just the distance from the center of the circle to the circle's edge. It is a property which serves to describe how big the circle is. The larger the radius, the larger the circle. In addition, you should also note that the object is moving around the circle at a constant speed, v. In the picture above, the direction of the object's motion is represented by the red arrow. As we discussed in a previous section, the direction of the motion is always perpendicular to the imaginary line connecting the object with the center of the circle. In other words, the direction of the object's velocity is tangent to the circle around which it is moving.

   As you will recall from a previous section, a force is responsible for making the object perform uniform circular motion. Given that an object is performing uniform circular motion (i.e., moving around a circle of radius, r, at a constant speed, v), the above formula tells us the amount of force responsible for causing the object to perform uniform circular motion. It is important to remember that it is this central pulling force, , which is causing the object to perform uniform circular motion. Not the other way around. Without this force, , the object would not be performing uniform circular motion. In fact, without any force, the object would be moving at a constant velocity (i.e., moving in a straight line at a constant speed), in accordance with Newton's 1st Law of Motion.

   Finally, the above formula solves for the amount of force needed to make an object undergo uniform circular motion. In other words, it solves for the magnitude of the force. It does not tell us the direction in which the force is applied. However, that's no big loss because we know the direction of this force must always point from the object toward the center of the circle it is traveling around. The subscript, c, in serves to remind us of this fact.

   Before proceeding, let me give you a warning about the above formula. It can only tell us the amount of force needed to make an object undergo uniform circular motion. It cannot tell us the nature or source of the force responsible for making the object undergo uniform circular motion. For example, if you swing a ball around overhead, the force responsible for making the ball undergo uniform circular motion is you pulling on the string which then pulls on the ball. However, if you were not given this information, the above formula would not be able to tell you this. To repeat, the above formula can only tell you the amount of force needed to make an object undergo uniform circular motion. If you want to know the nature of the force responsible, you will have to look elsewhere for that information. This is because there does not exist a new mysterious force whose sole purpose is to make things undergo uniform circular motion. Any force can make an object undergo uniform circular motion if it is capable of pulling an object inward toward the center of the circle around which it is traveling.

   Well, all this typing is making me jumpy, so let's do an example. It should make things more clear.

   • Example 1:
     Consider an object with a mass of 5 kilograms. In addition, the object is moving with a constant speed of 4 m/s around a circle of radius, 2 meters. What is the amount of force (in Newtons) which is causing this object to perform uniform circular motion? In addition, what is the direction in which this force is acting?

     This problem deals with uniform circular motion (i.e., an object moving at a constant speed around a circle) and asks for the amount of force that is causing this object to perform uniform circular motion. Well, the above formula is exactly what we need for this problem. All we have to do is assign the correct values to the appropriate variables. Try doing this on your own before looking below.

     • m = 5 kilograms
     • v = 4 m/s
     • r = 2 meters

     Notice that, because v is speed, we do not have to assign a direction to it. Now that we have all the variables assigned, we can just put them into the above formula, and the problem practically solves itself.

     

     

     

     Well, that was fairly simple. 40 Newtons is the amount of force (i.e., the magnitude of the force) needed to make the object move at a constant speed of 4 m/s around a circle with a radius of 2 meters. As for the direction of the force, we know that from our discussion in a previous section. The force is always pulling inward from the object to the center of the circle, regardless of the where the object is on the circle.

   So, you see, the difficult part of uniform circular motion was already tackled on the previous page. All we are doing here is filling in some details. Uh oh, my crystal ball tells me there is one .... no wait .... two thought questions coming up.

   • Thought Question:
     In example 1 above, does the inward force of 40 Newtons only act on the object at one particular point in the circle, or does this force need to be acting on the object all of the time?

     (Hint: Think about our old friend, Newton's 1st Law of Motion.)

     The answer is that the force of 40 Newtons must be on all the time, in order to make the object undergo uniform circular motion. Why is this true? Well, think about what happens when we shut off the 40 Newton force. According to Newton's 1st Law of Motion, as soon as we shut off the force, the object will then move at a constant velocity (i.e., moving in a straight line at a constant speed). In other words, as soon as we shut off the 40 Newton force, the object will no longer undergo uniform circular motion. Therefore, to make the object in example 1 undergo uniform circular motion, the 40 Newton force must always be on. Wherever the object is on the circle, a force of 40 Newtons must be acting on it pulling the object inward toward the center of the circle.

   • Thought Question:
     What is the nature of the force responsible for the uniform circular motion in example 1 above? Can this question even be answered given only the information found in example 1 above?

     Think about this for awhile. If you need a hint, refer to item #6 in this section.

     Well, this is sort of a trick question. It cannot really be answered. The only thing we can deduce from the example above is that there is a force of 40 Newtons pulling inward on the object toward the center of the circle around which it is moving. However, we cannot figure out the nature of the force solely from the information given in the example above. In other words, we are not given enough information (in the above example) to determine the nature of the force responsible for making the object undergo uniform circular motion.

     Now, if I were to add that the object in the example above was a ball attached to a string, then we would be able to answer this question. In this case, we could say the force responsible for the ball undergoing uniform circular motion is the person pulling on the string which then pulls on the ball causing it to undergo uniform circular motion. However, since this information was not given in the above example, we could not answer this thought question because we were not given enough information.

     So, why did I ask a question with no answer? Well, I wanted to reiterate the important point that the above formula can only tell us the amount of force (i.e., the magnitude of the force) needed to make an object undergo uniform circular motion. It cannot tell us the nature of the force responsible for making the object undergo uniform circular motion. To find the nature of the force, we need additional information.

   Now that we know the formula used to calculate the amount of force needed to make an object undergo uniform circular motion, let's see some of the factors upon which it depends.

   • Relationship between Force and Speed

     Consider the case of a ball attached to a string. Qualitatively, we know that the harder we pull on the string, the faster the ball spins around (assuming the radius of the circle is not changed). In fact, the above formula tells us how much harder to pull if we want to make the object, undergoing uniform circular motion, swing around faster. Let's rewrite the formula here again for convenience.

     

     In this section, we want to investigate how the required amount of force, , depends on how fast we want the object to move around the circle. Since we want to focus on the speed of the object and its relationship with the amount of force, we will change the speed of the object and then see how much force is required to make the object undergo uniform circular motion at the new speed, while keeping the mass of the object and the radius of the circle unchanged. By only adjusting the speed of the object, we are able to study its relationship with the required amount of force, , without the added distraction of the radius and mass.

     One word of caution before proceeding to the examples. While doing these examples, always keep in mind that it is the inward pulling force which is causing the object to undergo uniform circular motion. In addition, by adjusting the amount of this inward pulling force, we can adjust the speed with which the object moves around the circle. It is very important to keep this causal relationship in mind. It is wrong to say that increasing the speed of the object causes the force to be increased. However, it is correct to say that increasing the amount of force causes the object's speed to be increased. This is a slight but very important distinction.

     • Example 2:
       Consider three different objects, each having the same mass of 4 kg. In addition, assume that each object is undergoing uniform circular motion around its own circle with a radius of 2 meters. The first object is moving around its circle at a constant speed of 1 m/s. The second object is moving around its circle at a constant speed of 2 m/s. The third object is moving around its circle at a constant speed of 3 m/s. What is the amount of force, , needed to make the object undergo uniform circular in each of these three situations?

       First Object Speed v = 1 m/s

       Second Object Speed v = 2 m/s

       Third Object Speed v = 3 m/s

       
       So, you see, we have three almost identical situations. The mass of the object and the radius of the circle are the same in all three situations. The only difference is the speed of each object, undergoing uniform circular motion. By seeing how much force is required to make each object undergo uniform circular motion and then comparing the forces, we will be able to find a relationship between the force and the speed of an object undergoing uniform circular motion. Ok, back to the problem.

       This shouldn't be hard to do because we have already done an example above. However, what I want you to do is observe how the amount of force, , changes as the speed changes from the first object to the second object. Also, pay attention to how the amount of force changes from the first object to the third object. If you will notice, the speed of the second object is twice the speed of the first object, and the speed of the third object is three times the speed of the first object. In other words, I want you to see what happens to the amount of required force when we double and then triple the speed. Figure out the solutions on your own and then compare them to the answers below.

       Pull down to see the answers Solution for v = 1 m/s Solution for v = 2 m/s Solution for v = 3 m/s
       Solution: is the amount of force, , required.
       Well, the first thing you should notice is that the amount of force needed to make the object undergo uniform circular motion is different for all three objects. In particular, a greater amount of force is required to make an object undergo uniform circular motion at a higher speed. In other words, the harder the force pulls on the object undergoing uniform circular motion, the faster the object moves around the circle.

       However, we want to be a little more specific. In other words, how exactly does the amount of force, , change as we double and then triple the speed? Remember, the speed of the second object is double the speed of the first object, and the speed of the third object is triple the speed of the first object.

       Take some time to figure this out before looking at the answers below.

       Pull down to see the answers As the speed is doubled from 1 m/s to 2 m/s, As the speed is tripled from 1 m/s to 3 m/s,
       Solution:
       Do you notice a relationship between the speed and the amount of force required to make the object undergo uniform circular motion? Below, I've listed the results from this example.
       Results:
       • The second object's speed is 2 times that of the first object's speed. The inward pulling force acting on the second object is 4 times the force pulling on the first object. In other words, if the force pulling on the second object is 4 times the force pulling on the first object, the second object's speed will be greater than the speed of the first object by 2 times.

         If you will notice, 4 is just 2 times 2.

         4 = 2 x 2

       • The third object's speed is 3 times that of the first object's speed. The inward pulling force acting on the third object is 9 times the force pulling on the first object. In other words, if the force pulling on the third object is 9 times the force pulling on the first object, the third object's speed will be greater than the speed of the first object by 3 times.

         If you will notice, 9 is just 3 times 3.

         9 = 3 x 3

     So, what does this all mean? Well, for example, if we want to increase the speed of an object undergoing uniform circular motion by 4 times, all we have to do is increase the amount of force pulling on the object by 4 x 4 times. In other words, if we increase the force pulling on an object by 16 times, its speed will increase by 4 times. Keep in mind that this only applies to situations where the object's speed changes, while the mass and radius remain unchanged.

     In general, to increase the speed by a certain amount of times, we have to increase the inward pulling force by the square of the amount of times the speed of the object (undergoing uniform circular motion) is changed. Remember, when you square something, it just means to multiply it by itself. For instance, the square of 2 is 2 x 2, which is equal to 4. Once again, keep in mind that this conclusion only applies to situations where only the object's speed has changed, while the mass and radius remain unchanged.

     Now, that certainly was a lot of work that we just did to figure out the relationship between the object's speed and the amount of force required to make an object undergo uniform circular motion. However, is there an easier way to see this relationship without resorting to such extraordinary means?

     Well, we could have just looked at the formula above, which I will rewrite below.

     

     Looking at this formula, we can see the relationship between the amount of force, , and the speed, v, by isolating the term in the formula which depends solely on the speed. Since we assume the mass and radius remain unchanged, we can focus on just the term involving the speed.

     is related to the speed, v, by the following term:

     (where = v x v)

     
     By looking at the formula, we see that , the amount of force required to make an object undergo uniform circular motion, is related to the object's speed, v, by the term . Remember, is "v squared", which means v x v.

     Saying that the amount of force, , is related to the speed, v, by the term , is just a mathematical way of saying what we found in example 2 above. In other words, to increase the speed by a certain amount of times, we have to increase the inward pulling force by the square of the amount of times the speed of the object (undergoing uniform circular motion) is changed. For example, if we want to increase the speed of the object (undergoing uniform circular motion) by 6 times, we would have to increase the inward pulling force by 36 times. We get 36 from 6 squared, which is just 6 x 6. In other words, if we increase the amount of force pulling on an object (undergoing uniform circular motion) by 36 times, the object's speed will be increased by 6 times. Once again, these conclusions only hold true for an object (undergoing uniform circular motion) whose mass and radius remain unchanged. The speed can change because that is the property we are studying, but the radius and mass must remain unchanged.

     Well, I hope that last part made sense. If it didn't make sense, go back and review example 2 above again. The main point I wanted to get across was that the conclusion we arrived at in example 2 above could have been easily deduced by just looking at the formula above and picking out the factor which only depends on v, the speed of the object undergoing uniform circular motion. With practice, this will become second nature to you, and you won't need to resort to what we did in example 2 above.

     Now that we know the relationship between the object's speed and the amount of force required to make the object undergo uniform circular motion, let's do one example to solidify what we've just learned.

     • Example 3:
       Consider an object moving at a constant speed of 4 m/s around a circle with a radius of 2 meters. In other words, the object is undergoing uniform circular motion. How much must the inward force be increased if we want the object to move at 8 m/s without changing the radius of the circle around which the object moves?

       This example is a straightforward application of what we've just learned. Try it on your own before looking at the answer below. Notice that we don't need to know the object's mass to solve this problem.

       Pull down to see the answer In order to increase the speed from 4 m/s to 8 m/s, the force must be increased by 4 times.
       Here is how we arrived at the solution. We want to increase the speed of the object undergoing uniform circular motion from 4 m/s to 8 m/s, without changing the radius. In other words, we want to increase the object's speed by 2 times because 8 m/s is two times greater than 4 m/s. Once we get the problem to this stage, we can apply our conclusion from above. Since we want to increase the speed by 2 times, we need to increase the inward pulling force by 2 squared. In other words, we need to increase the inward pulling force by 4 times because 4 = 2 x 2.

   • Relationship between Force and Radius

     Now that we know the relationship between the object's speed and the amount of force required to make it undergo uniform circular motion, let's see what the relationship is between the amount of force and the radius of the circle around which the object moves. Because we want to study the relationship between the amount of force and the radius in this section, we will only adjust the radius of the circle around which the object undergoes uniform circular motion, while leaving the speed and mass unchanged.

     This time around, let's see if we can guess the relationship between the amount of force and the radius from the formula below. After doing so, we will use an example to verify our conclusion.

     

     To focus on the relationship between the amount of force and the radius while leaving the mass and speed unchanged, we can just look at the factor in the above formula that depends on the radius of the circle. Looking at the above formula, we arrive at the following conclusion.

     is related to the radius of the circle by the following factor:

     The factor,

     ,can be written as 1/r

     In other words, if we change the radius of the circle around which the object is undergoing uniform circular motion, 1/r tells us how the amount of required force, , is changed. By following the same line of reasoning we used to determine the relationship between the amount of force and the speed, we can determine the relationship between the amount of force and the radius.

     For example, let us consider an object undergoing uniform circular motion. While keeping the object's speed and mass unchanged, let us see how the amount of required force changes if the radius of the circle is changed. In particular, let's double the radius of the circle around which the object moves. In other words, let's make the radius of the new circle 2 times that of the old circle. Next, the factor, 1/r, tells us that if we increase the radius by 2 times, the amount of force will be changed by 1/2 times. In other words, the required amount of force will be halved.

     To repeat, if we want to increase the radius of the circle, around which the object undergoes uniform circular motion, by 2 times, the amount of force required, to make the object undergo uniform circular motion at the new radius, will be changed by 1/2 times. In other words, the amount of required force will be halved. Once again, this is assuming the speed of the object remains unchanged as the radius of the circle is increased.

     Similarly, the amount of required force is reduced by 1/3 if the radius of the circle, around which the object undergoes uniform circular motion, is increased by 3 times, while keeping the object's speed and mass unchanged.

     In general, if the object's speed and mass remain unchanged, the amount of force needed to make an object undergo uniform circular motion decreases as the radius of the circle, around which the object moves, increases. This is the conclusion we obtain from the above formula. However, is there a qualitative way to see that this should be true without doing any calculations?

     Well, think about the force which is causing the object to undergo uniform circular motion. As we know, this inward pulling force is responsible for turning the object around the circle at a constant speed. Now, as the radius of the circle increases, the curvature of the circle decreases because the circle gets larger. In other words, the larger the circle, the less the object needs to "turn" as it moves around the circle because the curvature of the circle is less. As a result, because the object doesn't need to "turn" as much as it moves around the larger circle, less force is needed to make the object "turn" around the larger circle because the larger circle has less curvature. In other words, less force is needed to "turn" an object around a circle with a larger radius. This is just a qualitative (and less accurate) way of saying that the amount of force needed to make an object undergo uniform circular motion is related to the radius of circle by the factor: 1/r. If you want an example to help make this concept believable, think about turning corners in your car. If you turn at the same constant speed in both corners, more force is needed to turn a car through a "sharp" small corner versus a larger corner.

     Now that we know the relationship between the radius of the circle and the amount of force needed to make an object undergo uniform circular motion, let's do an example. This example will demonstrate the conclusion we arrived at above.

     • Example 4:
       Consider two objects, each having a mass of 2 kg. In addition, assume both are undergoing uniform circular motion at a constant speed of 6 m/s. The first object moves around a circle with a radius of 2 meters. The second object moves around a circle with a radius of 4 meters. Find the amount of force which is making each object undergo uniform circular motion.

       First Object Radius r = 2 meters

       Second Object Radius r = 4 meters

       If you will notice, both objects are moving at the same speed around their respective circles. However, the second object is moving around a circle with a radius that is 2 times greater than the radius of the circle around which the first object is moving.

       Next, solve for the force required to make each object undergo uniform circular motion. This should be straightforward. Once again, try it on your own before looking at the answers below.

       Pull down to see the answers Solution for r = 2 meters Solution for r = 4 meters
       Solution: is the amount of force, , required.
       Next, let's see by how much the required amount of force has changed as the radius is doubled from 2 meters to 4 meters.

       As the radius is doubled from 2 meters to 4 meters, the amount of force (needed to make the object undergo uniform circular motion) is changed from 36 Newtons to 18 Newtons. As you can see, the required amount of force has been halved as the radius was doubled. In other words, if the speed and mass remain unchanged, the amount of force needed to make an object undergo uniform circular motion is reduced by half if the radius of the circle, around which the object moves, is doubled.

       If you will recall, this is precisely the result we obtained above by just looking at this formula and looking at the factor, 1/r, which depends solely on the radius of the circle around which the object moves. This shouldn't come as a surprise to anyone.

     This last example demonstrates the concept that , the amount of force required to make an object undergo uniform circular motion, is related to the radius of the circle by the factor, 1/r. Well, that's all for this section.

   • Summary

     In this section, we introduced the formula which describes how much force is required to make an object undergo uniform circular motion. I've written the formula again below.

     The primary purpose of this formula is to calculate the amount of force needed to make an object undergo uniform circular motion at a speed, v, while moving around a circle of radius, r. This formula does not tell us the direction in which the force must be applied. For that, we have to rely on our understanding of uniform circular motion from a previous section. As you will recall, the direction of the force always points from the object (undergoing uniform circular motion) toward the center of the circle around which it moves.

     In addition, the above formula only gives us a way to calculate the amount of force (i.e., the magnitude of force) required to make an object undergo uniform circular motion. It does not give us any information about the nature or source of the force. To figure out the nature of the force, we would have to be given additional information. For example, in the case of a ball attached to a string, the nature of the force, causing the ball to undergo uniform circular motion, is you pulling on the string which then pulls inward on the ball toward the center of the circle around which it is moving.

     While the previous page gave us a general understanding of uniform circular motion, this formula allowed us to be more specific. In particular, it allowed us to understand exactly how the required amount of force changes as we change the object's speed and the radius of the circle around which the object undergoes the uniform circular motion.

     Main Conclusions
     1. In general, if we want to increase the speed with which the object undergoes uniform circular motion (while keeping the radius unchanged), we need to increase the amount of force pulling inward on the object.

        Next, we showed how the above formula tells us exactly how much to change the amount of force in order to make the object's speed increase. In particular, the amount of force (required to make an object undergo uniform circular motion) is related to the object's speed by the factor below.

        is related to the speed, v, by the following term:

     2. In general, if the radius of the circle, around which the object undergoes uniform circular motion, is increased (while keeping the object's speed unchanged), the amount of force required to make the object undergo uniform circular motion (around the new circle with the larger radius) is decreased.

        Next, we showed how the above formula tells us exactly how much the amount of force changes as the radius of the circle changes. In particular, the amount of force (required to make an object undergo uniform circular motion) is related to the radius of the circle by the factor below.

        is related to the radius of the circle by the following factor:

     While the focus of this section was on uniform circular motion, we did something else of equal importance which will be useful in other sections as well. In this section, we looked at the formula we were given and, in a sense, "tore it apart". We investigated how the amount of force depends on various factors in the formula, namely the speed and radius. In addition, we saw that this information could be readily obtained by just looking at the formula and focusing on the relationship between the amount of force and the other factors: the speed and radius.

     For instance, we studied the relationship between the amount of force and the object's speed by isolating the factor in the formula that depends solely on the speed. After doing so, we looked at how a change in speed changes the required amount of force, while keeping all other properties, like the mass, unchanged.

     When you encounter a new formula, performing this sort of analysis is often very useful because, by doing so, you will get a better understanding of the formula and the physics behind it. If you will recall, we did a similar analysis in a previous section on gravity. It helped us to see what factors influenced the force of gravity. Likewise, by doing a similar analysis in this section, we were able to see the relationship between the object's speed and the amount of force required to make an object undergo uniform circular motion. In addition, we performed a similar analysis to see the relationship between the radius and the amount of required force. Doing this sort of analysis when you encounter a new formula is an important skill to learn. As with all things, it will get easier with practice. If you had some difficulty with the analysis in this section, take some time and reread the section. The time you spend on learning this type of analysis here will help you when we encounter new formulae in the future.