Degrees of Freedom Concepts - quick question.

[binbagsss]

So the number of degrees of freedom of a system is defined as the number of dynamical variables required , where the definition of dynamical variables is to completely descibe the configuration -positions of parts of a mechanical system.

The system I am to consider is a coin balanced on its edge and placed on a fixed, horizontal surface.

Questions:
- when considering what variables are required to specfify all parts of the system - which parts of the coin do we need to specify. i.e- surely we work with it's centre of mass . Do we model it is a particle? Otherwise how do you split the system into 'parts'. What makes us priorities one position of the coin over another?
- Is the table part of the system? This concept always confuses me ! Do we need to specify the position of the table?
- Do we neglect parametes required to descirbe , say, which way the coin is facing? And why? Does this depend upon the coin being identical in physical properties on both sides?(e.g even mass distribution each side)

- Some key things that we need to specify that I can think of include: distance along the table and position of the table.
(here I am assuming that the answer to the last part of question 2 is yes. Does the above sound reasonable?

Many thanks to anyone who can shed some light on this. Greatly appreciated ! :)

 

[Student100]

You're overthinking this. You can make the model as complicated or simple as you like. Does the horizontal surface play a role if it’s assumed to be uniform/unchanging? Does center of mass play a role in your system if you treat the coin as a point particle with two states? You could go on and on and on, it just depends on what you want your system to do and how well you know the initial values.

If your coin is heads in one moment of time, and tails in the next and oscillates between the two, how many degrees of freedom do you have? What’s important for the deterministic nature of the system?

The real question is, what do you want your system to do...? What's the question trying to be answered?

 

[voko]

The coin itself is a rigid body. How many degrees of freedom do you need to describe a rigid body?

Then, "balanced on its edge and placed on a fixed, horizontal surface" is a bit vague, but it means that there are some constraints on the coin. Constraints usually remove degrees of freedom. Analyze what your constraint really means and figure out how it modifies the original degrees of freedom.

Sometimes, we also have "natural" constraints. For example, when we analyze projectile motion, we use the fact that such motion always happens in a plane, so we use 2 coordinates to describe it, even though, in principle, the projectile has three degrees of freedom. Think whether your problem may have such natural constraints.

 

[haruspex]

Here's one way to think of it. I tell you I have balanced a coin on edge somewhere on a horizontal surface (on a square, say). You have a similar coin and square, and you are tasked with recreating my set-up exactly. You can ask me for a number of measurements, the answer to each being a single scalar. How many measurements do you need?
Bear in mind that a coin is not the same as a plain disc.

 

[binbagsss]

Here's one way to think of it. I tell you I have balanced a coin on edge somewhere on a horizontal surface (on a square, say). You have a similar coin and square, and you are tasked with recreating my set-up exactly. You can ask me for a number of measurements, the answer to each being a single scalar. How many measurements do you need?
Bear in mind that a coin is not the same as a plain disc.

Thanks, I did not realize the part 'you have a similar coin and square'.

To recreate the set up exactly, are the measurements needed : distance from, say, an edge of the table, and , it's orientation - i.e- laying flat / balanced on edge?So that's 1DOF (as we are already informed of it's orientation-on edge)

Regarding the coin is not the same as a plain disc-I believe this means in terms of heads/tails - that is, both sides are distinguishable. So this leads me to include another measurement - whether the coin is on heads/tails , or when it's on its edge, which way the heads faces. So I now have 2 DOF, does this justification sound reasonable?

Thanks.

 

[haruspex]

Thanks, I did not realize the part 'you have a similar coin and square'.

That is a bit of an assumption on may part, but I feel the coin and the platform are both givens, so you are only concerned with how the coin is placed on the platform.

To recreate the set up exactly, are the measurements needed : distance from, say, an edge of the table,

How many numbers does it take to specify the location of something within an area?

and , it's orientation - i.e- laying flat / balanced on edge?So that's 1DOF (as we are already informed of it's orientation-on edge)

As you say, you are given it is on edge, but there is still an orientation variable.

Regarding the coin is not the same as a plain disc-I believe this means in terms of heads/tails - that is, both sides are distinguishable. So this leads me to include another measurement - whether the coin is on heads/tails , or when it's on its edge, which way the heads faces. So I now have 2 DOF, does this justification sound reasonable?

Thanks.

The heads/tails aspect is taken care of by the orientation parameter you omitted. But if there is a design on the faces then there is another orientation parameter.

 

[binbagsss]

That is a bit of an assumption on may part, but I feel the coin and the platform are both givens, so you are only concerned with how the coin is placed on the platform.

How many numbers does it take to specify the location of something within an area?

As you say, you are given it is on edge, but there is still an orientation variable.

The heads/tails aspect is taken care of by the orientation parameter you omitted. But if there is a design on the faces then there is another orientation parameter.

2 parameters to specify location on an area.
Regarding the heads/tails aspect, is this what you meant by the coin is not the same as a plain disc?
Also, how is it that this aspect has been taken care of by the orientation parameter - is it because the coin is on its edge? If it was flat on the table, would you need to specify whether heads/tails is facing upward?

 

[voko]

If it was flat on the table, would you need to specify whether heads/tails is facing upward?

That would depend on two things: whether this is important; whether the coin can magically flip its sides.

Back to the coin on the edge. Can it roll with or without slipping? Can it move sideways?

 

[haruspex]

2 parameters to specify location on an area.

Yes.

Regarding the heads/tails aspect, is this what you meant by the coin is not the same as a plain disc?

No. You now have fixed the point on the horizontal surface where the coin makes contact.
What axes can you rotate the disc on without changing that?

 

[binbagsss]

That would depend on two things: whether this is important; whether the coin can magically flip its sides.

Back to the coin on the edge. Can it roll with or without slipping? Can it move sideways?

-In most cases this would not be important, and it can not change sides, but if it is important , and can't flip sides, wouldn't you need to specify the side heads/tails are facing initially?

It can not move sideways and it can roll without slipping?
-

 

[binbagsss]

Yes.
No. You now have fixed the point on the horizontal surface where the coin makes contact.
What axes can you rotate the disc on without changing that?

A vertical axis passing through this contact point and the centre of mass of the coin?

 

[voko]

-In most cases this would not be important, and it can not change sides, but if it is important , and can't flip sides, wouldn't you need to specify the side heads/tails are facing initially?

In classical mechanics, no, because that would not affect the equations of motion in any way.

In quantum mechanics, if the "face" was something like "spin", that would be necessary, because it affects dynamics.

It can not move sideways and it can roll without slipping?-

In this case the full dynamical description needs just one variable, so there is one degree of freedom. You would need more parameters to describe the static layout, but that is not what we understand by degrees of freedom.

 

[haruspex]

In this case the full dynamical description needs just one variable, so there is one degree of freedom. You would need more parameters to describe the static layout, but that is not what we understand by degrees of freedom.

We seem to be interpreting the OP quite differently. I base my interpretation on this line:

to completely describe the configuration -positions of parts of a mechanical system.

To me, that does mean the static layout.
There's nothing mentioned about friction, so even if you take a purely dynamic view the coin could slide around, leading to the same answer as the static analysis.

 

[voko]

Well, to me anyway, degrees of freedom are how many coordinates you need to describe the dynamics of a system in the simplest way possible. There are infinitely many parameters in any system, but a physical model has to select a reasonable number of them.

 

[haruspex]

A vertical axis passing through this contact point and the centre of mass of the coin?

Yes, but if there's a design on the faces of the coin then there's another of interest through the centre of the coin. Ignore friction.