Modern Physics- Degrees of freedom

[giraffe]

Homework Statement

i have a homework set to determine the degrees of freedom for certain objects. the professor never explained how to go about figuring these out and the book doesn't quite explain it well. two problems from the set are 1. a linear spring in 3space and 2. a bead constrained to move on a helix of constant pitch and constant radius

a third problem which i have no idea about is a compound pendulum and how many dof it has.

Homework Equations

equipartition of energy.

The Attempt at a Solution

i know there are 3 dof(degrees of freedom) for translational and rotational motions. so a total of 6. for the linear spring I am thinking that there has to be 3 plus 2 rotational because it is a linear object so total of 5. for the bead i think it might be 4 since both pitch and radius are constant. for compound pendulum, since each pendulum is only able to move up and down,left to right, than 2dof per each so a total of 4.

i do know there can be more then 6 dof depending on the objects or even molecules you might be dealing with. just looking for some clarification on how to think about these problems versus me just making an educated guess and not truly know why.

thanks for any help.

 

[jz92wjaz]

The degrees of freedom are the minimum number of variables you require to be able to fully describe the properties of a system.

The linear spring does not have fixed length, so you have to account for that as well. For the helix, how many variables do you need to uniquely identify the location of the bean on the helix? Hint: the bead has to be on the helix, so what is the smallest amount of information you can give me that would allow me to know exactly where on the helix it is?

For the compound pendulum, what is the smallest number of variables you can give me that would allow me to replicate your exact setup?

 

[giraffe]

Linear spring: it would have all 6 since it can be as big,long,small,short etc as it wants to.

Helix: up/down (vertically) 1 dof it's radius and pitch are constant so that's 2 dof it doesn't have. So 4.

Pendulum : it can only move left/right and up/down so only 2 dof. But there's 2 pendulums so total of 4 dof.

 

[jz92wjaz]

I presume you are using a book by liboff. The dumbbell example has the dumbbell length as being constant, and the length of the dumbbell isn't a degree of freedom in the example. The degrees of freedom are the location on the plane (x,y), and the rotation of the dumbbell, or theta, giving 3 degrees of freedom for a dumbbell on a 2D plane. A 2D plane could be rotated, but we don't count the rotation of the plane in our degrees of freedom, only the location of the dumbbell within the plane. This is an important distinction. Even if a particle can move in 3D space, it may be confined to move in something that resembles a 2D plane or 1D line (the plane and line could both be bent). In those cases, there are 2 and 1 degrees of freedom, respectively.

Linear spring: Can you list the 6 degrees of freedom?
Helix: The helix question only asks for the location of the bead on the helix, not for the location of the helix in space.
Pendulum: Depending on if you are swinging it in 3-d or 2-d space, you get different answers. If it were two dots on a 2D plane, there would be 4 degrees of freedom (x and y location for each dot). However, when those dots are constrained to move in a way where x and y are not independent, such as in the pendulum example, you have fewer degrees of freedom. You'll need to find variables that are independent of each other.

 

[giraffe]

Linear spring: the 3 translational & 3 rotational dof.

Helix: since it's just the bead than you only need 2 dof.

Pendulum: so in 3 space it would only add 1 dof (the rotation about vertical) so total 5 instead of 4 in 2 space.

 

[jz92wjaz]

Linear Spring: One of your dofs is incorrect.
Helix: This has two plausible answers, with one that I believe is very strongly preferred. What is your argument for it being 2 dofs? (what are your dofs?)
Pendulum: Before I try to explain this too much further, the book I have lists this problem as "a compound pendulum (two pendulums attached end to end)". I just wanted to clarify that we both have the same understanding of the problem.

Hint: You can describe the location of a single pendulum in 2D space (I am assuming it is attached to a fixed point) using Cartesian coordinates x and y, but this isn't the simplest way to do describe the location of the end of pendulum, and gives an incorrect number of dofs.

One more hint, from my textbook's problems #1.2 and #1.3:
1.2 Show that a particle constrained to move on a curve of any shape has one degree of freedom.
1.3 Show that a particle constrained to move on a surface of arbitrary shape has two degrees of freedom.

 

[jz92wjaz]

Several examples to hopefully help you understand the subject:

Examples of 1 degree of freedom:
1. A car on a roller coaster track. It can't leave the track of the coaster, so you can determine where it is using only the value of x, where x is the location along the track.
2. Two cars on a roller coaster, where the second is welded to the first. Even though there are two cars, because they are firmly attached, they act as one car and we know the location of the second car if we know the location of the first. The only coordinate (dof) we need in order to determine the location of both cars is x.

Examples of 2 degrees of freedom:
1. Two unconnected cars on a roller coaster. Car 1's location is determined by x1, and car 2's is x2, for a total of 2 dofs.
2. Two cars on a roller coaster connected by a spring. Car 1's location is determined by x, and, because x gives car 1's location, we can infer the location of car 2 by how far the spring is stretched. The coordinates (dofs) are: x (location of car 1) and l (current length of the spring).
3. A particle on the surface of a sphere. Its location on the sphere's surface can be determined by knowing only θ and ø.

Examples of 3 degrees of freedom:
1. The location of a single particle in 3D space (x, y, and z)
2. The location of Pac-man in the old pac-man games. Pac-man has an x and y coordinate, and θ is used to determine what direction he is facing. Coordinates are x, y, and θ.

 

[giraffe]

I'm using a book called modern physics by Krane third edition.

Thanks for those examples as it help clear some stuff.

Helix- I thought two but after seeing those examples it should be 1.

Pendulum- that's the correct definition given for it to be compound. The end would only be 1 dof since it's fixed.

Linear spring - there are no restrictions/constraints to this one. So it's 6 since it can be any where.

 

[jz92wjaz]

Your spring coordinates are different than I was thinking (location of one end has 3 degrees, location of the other has three degrees, is my understanding of what you said), but work just fine. Helix looks good too.

The first pendulum (in the 2D scenario) would have 1 dof, whether or not a second pendulum is attached to it. How many additional dofs do you need to describe the current location of the second pendulum?

 

[giraffe]

Just 1 more. You could use an angle in relation to the first or use just an x value.