Why is the force of the spring equal to the centripetal force?

In summary, the potential energy stored in the spring is equal to the centripetal force multiplied by the radius of the rotating spring.
  • #1
Perseverence
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Homework Statement


A string is rotated around a point with a radius of 4 meters. Calculate the potential energy stored in the spring

Homework Equations


F = KX

The Attempt at a Solution


The solution to the problem involves making the force of a spring equal to the centripetal force. I don't really understand why that would be. Both forces Point towards the center of the circle. How do we know that they are equal? They are not counterbalancing each other if they are both going in the same direction.
 
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  • #2
But you know that's not the case by experiment as the string stretches and the object moves around in a circular path. Hence while the centripetal force is directed inward, you must look more carefully at the direction of the string tensile force.

http://en.citizendium.org/wiki/Centripetal_force

The figure shows the radially inward centripetal force FC provided by the string that is necessary to force the object to travel the circular path of radius R at a constant speed v. (The "speed" of the body is the magnitude of its velocity, without regard to direction.) The body in turn provides an equal but oppositely directed tensile force FT on the string as shown to the left (a consequence of Newton's law that to every force there is an equal and opposite reaction force), and the string also is subject to the same tensile force at its other end, in opposite direction, as provided by the fixed centerpost.
 
  • #3
It's a common confusion regarding the centripetal force.
It's not a real force - it's just a value some other force has to have in order to make a mass go around in circles.
So, if you have a mass on a string (or spring) going in circles, then you know that the net radial force is precisely equal to the centripetal force.
 
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  • #4
Perseverence said:

Homework Statement


A spring is rotated around a point with a radius of 4 meters. Calculate the potential energy stored in the spring.
Just to check you have the right picture in mind: how long is the rotating spring ?
 
  • #5
Bandersnatch said:
It's a common confusion regarding the centripetal force.
It's not a real force - it's just a value some other force has to have in order to make a mass go around in circles.
So, if you have a mass on a string (or spring) going in circles, then you know that the net radial force is precisely equal to the centripetal force.

Some other force or forces. Sometimes several forces combine to provide the centripetal force needed to make the mass move in a circle.
 
  • #6
Bandersnatch said:
It's a common confusion regarding the centripetal force.
It's not a real force - it's just a value some other force has to have in order to make a mass go around in circles.
It's probably better to say the centripetal force is not yet another force; rather, it's a resultant force that points radially inward. I wouldn't say it's not a real force, as that might cause confusion with the concept of a fictitious force, like the centrifugal force.

I usually avoid the idea of a centripetal force altogether. When an object follows a circular path, it experiences a centripetal acceleration, ##a_r = \frac{v^2}{r}.## It shows up on the ##ma## side of the second law, not on the ##\sum F## side.
 
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  • #7
vela said:
I usually avoid the idea of a centripetal force altogether. When an object follows a circular path, it experiences a centripetal acceleration, ##a_r = \frac{v^2}{r}.## It shows up on the ##ma## side of the second law, not on the ##\sum F## side.
I like that. So the tension in the string/rope/etc. is equal to the mass times the centripetal acceleration.
 
  • #8
I mostly agree with @vela in that "centripetal" is an adjective that indicates the direction of a force relative to some coordinate system, but I wouldn't necessarily call it the resultant. "Centripetal" is used the same way as the adjectives "normal", "horizontal", vertical", etc. to label the direction of forces. For example, the Earth exerts a vertical force on a book that is at rest on a table top; the Earth exerts a centripetal force on the book when it is in a circular orbit around the Earth. Same Earth, same book, same interaction between the two but different reference frames.
 
  • #9

1. Why is the force of the spring equal to the centripetal force?

The force of the spring and the centripetal force are equal because they both act in opposite directions to balance each other out. The force of the spring pulls the object towards the center while the centripetal force pushes the object away from the center. This creates a state of equilibrium where the object stays in a circular motion.

2. How does the force of the spring contribute to circular motion?

When an object attached to a spring is moving in a circular path, the spring will stretch or compress depending on the direction of the motion. This stretching or compression creates a force, known as the force of the spring, that acts towards the center of the circle. This force helps to maintain the circular motion of the object.

3. Is the force of the spring always equal to the centripetal force?

Yes, the force of the spring and the centripetal force are always equal in magnitude but act in opposite directions. This is known as Hooke's Law, which states that the force exerted by a spring is directly proportional to the distance it is stretched or compressed.

4. What happens to the force of the spring when the circular motion stops?

When the circular motion stops, the force of the spring also disappears. This is because the force of the spring is only present when there is a circular motion, which creates a change in the spring's length. When the motion stops, the spring returns to its original length and the force of the spring is no longer needed.

5. Can the force of the spring be greater than the centripetal force?

No, the force of the spring and the centripetal force are always equal in magnitude but act in opposite directions. If the force of the spring were greater than the centripetal force, the object would move towards the center of the circle, making the circular motion unstable.

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