UCM - Derive speed, of a body moving, radius r, and the frequency f.

In summary, the relationship between the speed v and frequency f of a body moving in a circle of radius r is v = 2πrf. This can be derived by substituting T=1/f into the formula v=2πr/T and simplifying. Additionally, the relationship between centripetal acceleration and frequency can be used to derive this relationship.
  • #1
MohammadG
17
0

Homework Statement


Derive a relationship between the speed v, of a body moving in a circle of radius r, and the frequency f, of the revolutions.

Homework Equations


v=2πr/T
T=1/F

The Attempt at a Solution



Well, I thought this would be as simple as solving for F in the second equations and substituting into the original formula.
Eg:

v=2πr/T
T=1/F
F=1/T

v=2πr/1/F

v=2πr/F

Is the above correct? Could I expand it into more steps, like how one would derive F = 1/T, I already know how to derive v=2πr/T.

Thanks!
 
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  • #2
MohammadG said:

Homework Statement


Derive a relationship between the speed v, of a body moving in a circle of radius r, and the frequency f, of the revolutions.

Homework Equations


v=2πr/T
T=1/F

The Attempt at a Solution



Well, I thought this would be as simple as solving for F in the second equations and substituting into the original formula.
Eg:

v=2πr/T
T=1/F
F=1/T

v=2πr/1/F

v=2πr/F

Is the above correct? Could I expand it into more steps, like how one would derive F = 1/T, I already know how to derive v=2πr/T.

Thanks!

Does not the T = 1/f follow from the way they are defined?

Centripetal acceleration is given alternately by mv2/R or 4∏2R/T2 which means 4∏2Rf2
by equating the first and last form I think the answer falls out.
 

1. What is UCM?

UCM stands for Uniform Circular Motion. It is a type of motion in which an object moves in a circular path at a constant speed.

2. How do you derive the speed of a body in UCM?

The speed of a body in UCM can be derived using the formula v = 2πrf, where v is the speed, r is the radius of the circular path, and f is the frequency of the motion.

3. Can the speed of a body in UCM change?

No, the speed of a body in UCM remains constant throughout the motion. This is because the radius and frequency of the motion remain unchanged.

4. How does the radius of the circular path affect the speed of a body in UCM?

The larger the radius of the circular path, the higher the speed of the body will be. This is because the distance traveled in one complete revolution increases with the radius, resulting in a higher speed.

5. What is the relationship between frequency and speed in UCM?

There is an inverse relationship between frequency and speed in UCM. As the frequency increases, the speed decreases, and vice versa. This is because the time taken for one complete revolution remains constant, so a higher frequency means a shorter distance traveled, resulting in a lower speed.

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