What distance is needed for a body to spin in circular motion with a spring?

[Eitan Levy]

Homework Statement

http://[URL=http://www.siz.co.il/][PLAIN]http://up416.siz.co.il/up3/w2jymjm5nqxy.png [/B]
A body with a mass that equals to m is inside a smooth tube and is attached to two identical springs with constants that equal to k. Their length when they are limp is L₀. When the body is in the middle of the tube both springs are limp.
The tube is attached to a pole that spins around with an angular velocity that equals to ω (The motion is only horizontal)
1. In what distance the body needs to be from the center of the tube so he will spin around along with the tube without moving in relative to him?
Answer: mω²L₀/(2k-mω²)
2. Now we attach the tube in an angle of 37 degrees to the pole (α=37 degrees), what the angular velocity ω₀ needs to be so both springs will remain limp?
2b. The angular velocity in increased to 2ω₀, in what distance the body needs to be from the center of the tube so he will spin around along with the tube without moving in relative to him?

Honestly I tried a lot but I can't seem to solve the first question, can anyone help please?
Sorry for my English, I know it's very poor.
Equations:
ma_r=F
a_r=ω²r
F=KΔL
ma=F

How I tried to solve this:
I tried to say that the force that the springs enables on the body is 2KΔL, but it didn't work.

 

[PeroK]

Can you start by calculating the centripetal force required for circular motion of the mass $m$?

 

[Eitan Levy]

Honestly I have know idea how to do it, we have just started to learn this topic.
Can you post a solution to the first question and I will try to understand from it?

 

[PeroK]

Honestly I have know idea how to do it, we have just started to learn this topic.
Can you post a solution to the first question and I will try to understand from it?

The purpose of this forum is to help you do problems that you perhaps can't quite do yourself. We assume you have some familiarity with the material.

The best thing to do is to find your notes on circular motion and centripetal force. There is also lots of material online about this subject. For example:

http://www.physicsclassroom.com/class/circles

You can always ask specific questions here if there is something in particular you don't understand.

 

[Buffu]

Honestly I have know idea how to do it, we have just started to learn this topic.
Can you post a solution to the first question and I will try to understand from it?

centripetal force is $mr\omega^2$ where r - radius, m - mass, $\omega$ - angular velocity.

 

[Eitan Levy]

The purpose of this forum is to help you do problems that you perhaps can't quite do yourself. We assume you have some familiarity with the material.

The best thing to do is to find your notes on circular motion and centripetal force. There is also lots of material online about this subject. For example:

http://www.physicsclassroom.com/class/circles

You can always ask specific questions here if there is something in particular you don't understand.

Sorry, I didn't explain myself quite well. We did work on this subject for a bit, but this question is much harder than what we did so far. I know the basics but this question is really hard for me.

 

[PeroK]

Sorry, I didn't explain myself quite well. We did work on this subject for a bit, but this question is much harder than what we did so far. I know the basics but this question is really hard for me.

You appear to be stuck on the basics. @Buffu has given you the answer to my first question above.

 

[Buffu]

1. In what distance the body needs to be from the center of the tube so he will spin around along with the tube without moving in relative to him?

You just need to find the displacement of the object inside the tube due to its rotation.
Forget everything else written in the question especially "the tube without moving in relative to him".

 

[Buffu]

Sorry, I didn't explain myself quite well. We did work on this subject for a bit, but this question is much harder than what we did so far. I know the basics but this question is really hard for me.

You should consider the object when all the forces acting on it will be at equilibrium.
So first write out all the forces acting on the body.

 

[Eitan Levy]

You should consider the object when all the forces acting on it will be at equilibrium.
So first write out all the forces acting on the body.

This I realized, I tried to do it but I didn't get the correct answer. Can you explain the forces in the question please, obviously gravity and normal, but to which direction the springs enable the force on the body, and what is the size of those forces?

 

[Eitan Levy]

You should consider the object when all the forces acting on it will be at equilibrium.
So first write out all the forces acting on the body.

Also, when I try this I don't get the angular velocity in any of the forces, and it appears in the final answer twice.

 

[Buffu]

Also, when I try this I don't get the angular velocity in any of the forces, and it appears in the final answer twice.

What answer did you get ?
Show the gist of what you did to get it.

 

[Eitan Levy]

What answer did you get ?
Show the gist of what you did to get it.

I said the left string enables a force that equals to kΔl=k(l₀+x) (x is the distance from the center), then I said the right string enables a force that equals to kΔl2=k(l₀-x), I also think both are in the same direction, but then they won't be equal to zero, if they are not and I want to check when they are equal I get an answer that is not even close. It looks like I am really missing something here, please, can you help?

 

[Buffu]

I said the left string enables a force that equals to kΔl=k(l₀+x) (x is the distance from the center), then I said the right string enables a force that equals to kΔl2=k(l₀-x), I also think both are in the same direction, but then they won't be equal to zero, if they are not and I want to check when they are equal I get an answer that is not even close. It looks like I am really missing something here, please, can you help?

First problem is that both of them won't be in same direction. Secondly you did not mention centrifugal force on the object.
The horizontal force on the object are :-
1) force by left spring that is toward the center of axis of rotation.
2) force by left spring that is away the center of axis of rotation.
3) centrifugal force (Find its direction)

Now can you equate them and get the answer ...

 

[Eitan Levy]

First problem is that both of them won't be in same direction. Secondly you did not mention centrifugal force on the object.
The horizontal force on the object are :-
1) force by left spring that is toward the center of axis of rotation.
2) force by left spring that is away the center of axis of rotation.
3) centrifugal force (Find its direction)

Now can you equate them and get the answer ...

Obviously it will be toward the center, but I still don't get the right answer. The equation isn't mω²r+kΔl1-kΔl2=0 and Δl1=-x and Δl2=x?

 

[Eitan Levy]

Anyone?

 

[haruspex]

The equation isn't mω²r+kΔl1-kΔl2=0 and Δl1=-x and Δl2=x?

yes, but what do you have for r there?
Please do not just say "I did not get the right answer". Post all your working so that we can see exactly what you did. We are not mind readers.

 

[Buffu]

Obviously it will be toward the center

Obviously you are wrong, It is not towards the center, It is away from center.

 

[Eitan Levy]

Obviously you are wrong, It is not towards the center, It is away from center.

Okay I understand that a centrifugal force will be directed away from the center, but when the force is centifugal and when it's centripetal? I was taught that the body is able to hae a circual motion thanks to the centripetal force, how there is a circual motion without it and how to know if the force is centrifugal or centripetal? We didn't discuss about this in class at and that's the last thing I don't understand here. Thanks in advance, you helped me a lot.

 

[PeroK]

Okay I understand that a centrifugal force will be directed away from the center, but when the force is centifugal and when it's centripetal? I was taught that the body is able to hae a circual motion thanks to the centripetal force, how there is a circual motion without it and how to know if the force is centrifugal or centripetal? We didn't discuss about this in class at and that's the last thing I don't understand here. Thanks in advance, you helped me a lot.

If the mass is displaced outwards, then both springs will apply a force on it inwards, creating the necessary centripetal force.

 

[Eitan Levy]

If the mass is displaced outwards, then both springs will apply a force on it inwards, creating the necessary centripetal force.

Okay, now I really need your help, I got a really long answer for 2b when the one in the book is very short.

You can't really see mg in the picture but those are the forces I used.
My equations were:
r=sinα(l₀+x)
Nsinα=2kxcosα+mg
4mω₀²r=Ncosα+2kxsinα

Now I put numbers in my final answer for x and the answer value was the same as in the book, but in the book the answer is:x=3mgcosα/(2k-4mgl₀cosα) and mine is very long. Are my forces and equations right? Thanks a lot.

 

[Buffu]

Your FBD is not clear. I maybe missing something but where does that 2kx come from ?

 

[haruspex]

when the force is centifugal and when it's centripetal?

Centrifugal and centripetal are two ways of viewing the same thing. You can choose either but don't mix them.
Centripetal is used when your reference frame is inertial. It is not an applied force. It is that component of the net applied force necessary to achieve the curvilinear motion required or observed.
Centrifugal force is a virtual, or "fictitious" force. It arises when you use the object's frame of reference. You may feel pinned to the wall by centrifugal force in a rotating drum. You feel the reaction force of the drum against you, yet in your reference frame you are stationary, so your brain invents a force pushing you against the drum to account for this.

For a longer discussion see https://www.physicsforums.com/insights/frequently-made-errors-pseudo-resultant-forces/

 

[Eitan Levy]

Your FBD is not clear. I maybe missing something but where does that 2kx come from ?

From the springs.

 

[haruspex]

Okay, now I really need your help, I got a really long answer for 2b when the one in the book is very short.

You can't really see mg in the picture but those are the forces I used.
My equations were:
r=sinα(l₀+x)
Nsinα=2kxcosα+mg
4mω₀²r=Ncosα+2kxsinα

Now I put numbers in my final answer for x and the answer value was the same as in the book, but in the book the answer is:x=3mgcosα/(2k-4mgl₀cosα) and mine is very long. Are my forces and equations right? Thanks a lot.

Your equations look fine, but I do not know what you have for ω₀. I cannot see a post where you finish answering 2a.

 

[Eitan Levy]

Centrifugal and centripetal are two ways of viewing the same thing. You can choose either but don't mix them.
Centripetal is used when your reference frame is inertial. It is not an applied force. It is that component of the net applied force necessary to achieve the curvilinear motion required or observed.
Centrifugal force is a virtual, or "fictitious" force. It arises when you use the object's frame of reference. You may feel pinned to the wall by centrifugal force in a rotating drum. You feel the reaction force of the drum against you, yet in your reference frame you are stationary, so your brain invents a force pushing you against the drum to account for this.

For a longer discussion see https://www.physicsforums.com/insights/frequently-made-errors-pseudo-resultant-forces/

Thanks a lot!

 

[Eitan Levy]

Your equations look fine, but I do not know what you have for ω₀. I cannot see a post where you finish answering 2a.

Now it hit me, I probably should use what I got in 2a. I am pretty sure it will be correct if you say the equations are fine. You all helped me to understand this topic much better, thank you all very much