# Circular motion of a rotational spring

• Ariano AnnaG
In summary: I tried to calculate the angular frequency for simple harmonic motion as if the toy were being oscillated on the spring. That's not relevant to this problem.
Ariano AnnaG

## Homework Statement

A child is playing with a spring (k=100000 N/m, Li = 0.5 m). One of his toys (m=0.5 kg) is attached to the further extremity. The child is rotating the spring above his head on a horizontal plane, with a uniform circular motion.

What is the elongation of the spring?
I’m not sure what is the right approach to this problem. I tried drawing a scheme of the forces as shown in the picture attached but I still have several doubts.
I thought the elastic force of the spring (Fel) that draws the mass towards the center, should be equal to the Centrifugal force (Fc) acting on the opposite direction and keeping the string in tension.

Hence I solved the problem equalizing them, as shown attached. I end up with a quite unlikely result (83.5m).
I don’t know if the initial condition is appropriate or if there are any other/different forces acting on the mass.

## Homework Equations

in the attached photo

## The Attempt at a Solution

in the attached photo

#### Attachments

• 562d53c5-22bb-435e-837b-3e2dbea06e99 copia.jpeg
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Ariano AnnaG said:
A child is playing with a spring (k=100000 N/m, Li = 0.5 m). One of his toys (m=0.5 kg) is attached to the further extremity. The child is rotating the spring above his head on a horizontal plane, with a uniform circular motion.
Looks like some info is missing from the problem statement. How fast is the toy being rotated?

Once you have all the data, you'll want to draw a free body diagram for the toy. (Two forces act on it.)

Ariano AnnaG
Ariano AnnaG said:
The attempt at a solution
Your attempt includes a calculation of the SHM frequency for oscillations of the spring, but the question statement implies the spring extension is constant, so this is irrelevant.
As @Doc Al says, there is missing info... could be e.g. the angle the spring makes to the horizontal.

Last edited:
Ariano AnnaG
Doc Al said:
Looks like some info is missing from the problem statement. How fast is the toy being rotated?

Once you have all the data, you'll want to draw a free body diagram for the toy. (Two forces act on it.)

the diagram of the forces acting on the toy is in the photo attached, it's quite small, between the datas and the drawing of the system.
I tought it could have been solved using angular speed ( radq(k/m) )
In the problem there isn't any other data

Ariano AnnaG said:
the diagram of the forces acting on the toy is in the photo attached, it's quite small, between the datas and the drawing of the system.
Yes, I saw that diagram but do not understand it. What two forces act on the toy? Those are the only forces that should appear in a free body diagram.

Ariano AnnaG said:
I tought it could have been solved using angular speed ( radq(k/m) )
Are you given the angular speed?

Ariano AnnaG said:
In the problem there isn't any other data
Well, I trust you realize that the "harder" the toy is swung, the more the spring will elongate. Thus, additional info is needed. Did you post the problem word for word?

Ariano AnnaG
you have k and m, so you can find angular speed, and I reported the problem word for word in the first post.
Here's the diagram of the forces
Ariano AnnaG said:

## Homework Statement

A child is playing with a spring (k=100000 N/m, Li = 0.5 m). One of his toys (m=0.5 kg) is attached to the further extremity. The child is rotating the spring above his head on a horizontal plane, with a uniform circular motion.

What is the elongation of the spring?
I’m not sure what is the right approach to this problem. I tried drawing a scheme of the forces as shown in the picture attached but I still have several doubts.
I thought the elastic force of the spring (Fel) that draws the mass towards the center, should be equal to the Centrifugal force (Fc) acting on the opposite direction and keeping the string in tension.

Hence I solved the problem equalizing them, as shown attached. I end up with a quite unlikely result (83.5m).
I don’t know if the initial condition is appropriate or if there are any other/different forces acting on the mass.

## Homework Equations

in the attached photo

## The Attempt at a Solution

in the attached photo

Ariano AnnaG said:
you have k and m, so you can find angular speed,
As @haruspex points out, you attempted to calculate the angular frequency for simple harmonic motion as if the toy were being oscillated on the spring. That's not relevant to this problem.
Ariano AnnaG said:
Here's the diagram of the forces
Again, I saw your diagram, but I don't understand/agree with it. Draw a new diagram in a vertical plane. What two forces act on the toy?

Ariano AnnaG
Thanks for the replies. I think you are correct. There is indeed a missing data which I guess should be the angular speed.
As haruspex pointed out, the angular speed I used is relative to the harmonic motion of a spring and thus irrelevant in this case.
However, given the angular speed, would the equation I used be correct?

Ariano AnnaG said:
However, given the angular speed, would the equation I used be correct?
No. You assumed that the centripetal force was equal to the spring force. Note: Is the spring horizontal?

Ariano AnnaG
Yes, the spring is horizontal.
For the elongation of the spring to be constant, shouldn't the sum of the forces at the level of the mass be 0?
Otherwise shouldn't the spring describe also an oscillation of harmonic motion while rotating?
That is the reasoning I made to equalise centrifugal force and spring force.

Ariano AnnaG said:
Yes, the spring is horizontal.
No. The toy moves in a horizontal plane, but the spring is not horizontal. This is why I keep asking you to draw a new free body diagram!

Ariano AnnaG said:
For the elongation of the spring to be constant, shouldn't the sum of the forces at the level of the mass be 0?
No. The mass is accelerating, thus there must be a net force on it in the direction of the acceleration. (Note: "centripetal force" is not a kind of force, it's just a description of the net force needed to create a centripetal acceleration.)

I made a lot of mistakes and the problem is phrased in an ambiguous way and with missing data (angular velocity ω). However now I think I figured it out a bit better. Probably what created most of my confusion is the fact that the children is rotating the object above his head. However, I think I could consider the system just like the spring was fixed to a horizontal plane without drag. Given this, the mistake in my free body diagram is drawing 2 forces when there’s only one acting on the mass in a relevant way for the problem: the centripetal force given by the elastic force exerted by the spring.Thus given that mω2r = -kΔl I can find the elongation expressing the radius as a function of ΔlDo you think this resolution may be correct?Thank you so much for your replies, and I beg you pardon again for my confusion.

Ariano AnnaG said:
I could consider the system just like the spring was fixed to a horizontal plane
It is unlikely the problem is intended that way, and it does not help in resolving the missing information.
Just solve the problem in a general way: relate spring constant k, relaxed length Li, angular velocity ω to radius of rotation, angle θ of spring to horizontal and spring extension.

## 1. What is meant by circular motion of a rotational spring?

The circular motion of a rotational spring refers to the movement of a spring when it is rotated around a fixed axis, causing the spring to stretch or compress in a circular pattern.

## 2. How does a rotational spring create circular motion?

A rotational spring creates circular motion through the force of torque, which is generated when a force is applied perpendicular to the rotational axis. This force causes the spring to rotate and create circular motion.

## 3. What factors affect the circular motion of a rotational spring?

The circular motion of a rotational spring can be affected by a variety of factors, including the strength and elasticity of the spring, the amount of torque applied, and any external forces acting on the spring.

## 4. How is the frequency of circular motion calculated for a rotational spring?

The frequency of circular motion for a rotational spring is calculated by dividing the spring's stiffness (measured in Newtons per meter) by its mass (measured in kilograms). This gives the frequency in units of hertz (Hz).

## 5. What are some real-life applications of circular motion of a rotational spring?

The circular motion of a rotational spring has many practical applications, such as in car suspension systems, door hinges, and mechanical clocks. It is also used in experiments to study rotational motion and in the construction of amusement park rides.

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