Radius as a function of Uniform Circular Motion

So it's really just a matter of perspective - whether you are looking at how the centripetal acceleration changes with radius, or how it changes with velocity.In summary, the relationship between radius and centripetal acceleration is that they are directly proportional, as shown in the formula a=(w^2)*r. However, when looking at the formula a=v^2/r, it may seem like increasing r would result in a decrease in acceleration. This is because the velocity, v, also depends on r, meaning that as r increases, so does v. Therefore, both formulas ultimately show the same relationship between radius and acceleration. It is just a matter of perspective, depending on if you are looking at the change in acceleration with respect to
  • #1
adam.kumayl
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How exactly does radius affect centripetal acceleration? In one formula we have (a=v^2/r) which implies inversely proportional, while in the other we have (a=(w^2)*r) which implies directly proportional. I understand that increasing r increases velocity (v=w*r) which means the right answer is increasing r increases acceleration, however how do we justify that increasing r would decrease a in the formula (a=v^2/r)?

Also how can I intuitively makes sense of this? Does increasing r not mean that the same velocity has more time to change (as Khan Academy states at 3:29-4:32) -->

This MIT lecture at 5:40-6:21, states that the acceleration would decrease as radius decreases.
http://www.youtube.com/watch?v=Otmg0-knGtE&list=ECF688ECB2FF119649

So which is it ? They seem to be saying opposite things. Am I misunderstanding something?

p.s. this site is amazing that people contribute their time and knowledge for free to help someone else they will never meet in their life. Truly thank you from one sentient being to another.
 
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  • #2
well, it depends if you want to hold omega fixed, or velocity fixed. The idea is that the centripetal acceleration must be a specific value, so that a perfect circle is made. So to keep to that requirement, if you have a larger radius, you must have a smaller omega. And equivalently, you must have a larger velocity.
 
  • #3
adam.kumayl said:
How exactly does radius affect centripetal acceleration? In one formula we have (a=v^2/r) which implies inversely proportional, while in the other we have (a=(w^2)*r) which implies directly proportional. I understand that increasing r increases velocity (v=w*r) which means the right answer is increasing r increases acceleration, however how do we justify that increasing r would decrease a in the formula (a=v^2/r)?

Also how can I intuitively makes sense of this? Does increasing r not mean that the same velocity has more time to change (as Khan Academy states at 3:29-4:32) -->

This MIT lecture at 5:40-6:21, states that the acceleration would decrease as radius decreases.
http://www.youtube.com/watch?v=Otmg0-knGtE&list=ECF688ECB2FF119649

So which is it ? They seem to be saying opposite things. Am I misunderstanding something?

p.s. this site is amazing that people contribute their time and knowledge for free to help someone else they will never meet in their life. Truly thank you from one sentient being to another.


So you've got a rotating disk, and you're interesting in the centripetal acceleration acting on a point on the disk at distance r from the centre of rotation.

For a given angular rotational frequency, the centripetal acceleration acting on the point is proportional to r, as your second equation says.

The reason your first equation seems to disagree is it because it contains v, the velocity of the specific point you are looking at. But the velocity of that point also depends on its distance from r, from the equation v = wr. So when you make r bigger, you are also making v bigger.

Substitute v=wr into your first equation and you will see that both are saying the same thing.
 
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FAQ: Radius as a function of Uniform Circular Motion

What is uniform circular motion?

Uniform circular motion is the motion of an object traveling in a circular path at a constant speed. This means that the object covers equal distances in equal periods of time and its velocity is always tangential to the circular path.

How is radius related to uniform circular motion?

The radius of a circle is the distance from the center to the edge of the circle. In uniform circular motion, the radius is the distance between the center of the circle and the object moving in a circular path. It determines the size of the circle and is directly proportional to the speed of the object.

What is the formula for calculating radius in uniform circular motion?

The formula for calculating the radius in uniform circular motion is r = v^2/a, where r is the radius, v is the velocity of the object, and a is the centripetal acceleration. This formula is derived from the relationship between centripetal force, mass, and velocity.

Does the radius affect the speed of an object in uniform circular motion?

Yes, the radius does affect the speed of an object in uniform circular motion. According to the formula for calculating radius, as the radius increases, the speed of the object also increases. This is because a larger radius requires a larger centripetal force to maintain the circular motion, resulting in a higher velocity.

What are some real-life examples of uniform circular motion?

Some real-life examples of uniform circular motion include the motion of planets around the sun, the movement of a Ferris wheel, and the rotation of a ceiling fan. These all involve objects moving in a circular path at a constant speed.

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