How Many Degrees of Freedom Are in These Systems?

[sudipmaity]

Homework Statement

Find the number of degrees of freedom for:
1. A diatomic gas molecule in an enclosure with constant interatomic distance.
2. Two particles constrained to move on a plane connected by a massless spring.

Homework Equations

Nn-k
Where n=no. of dimensions.
N=no.of particles.
k=no.of constraints

The Attempt at a Solution

1.N=2, n=3, k=1
So DOF=3*2 -1 =5
2.The particles are free to move on a plane.So if x describes the posn. y and z should be fixed.
So here n=1, N=2, k=0 (spring is massless).
So DOF=1*2-0=2.
Am i right??

 

[ehild]

Homework Statement

Find the number of degrees of freedom for:
1. A diatomic gas molecule in an enclosure with constant interatomic distance.
2. Two particles constrained to move on a plane connected by a massless spring.

Homework Equations

Nn-k
Where n=no. of dimensions.
N=no.of particles.
k=no.of constraints

The Attempt at a Solution

1.N=2, n=3, k=1
So DOF=3*2 -1 =5

Right.

2.The particles are free to move on a plane.So if x describes the posn. y and z should be fixed.

The position on a plane is given by two coordinates. The plane is two-dimensional, n=2

ehild

 

[sudipmaity]

Sorry. I guess it should be then DOF =4 for 2nd problem.

 

[ehild]

It is :)

 

[SalfordPhysics]

Correct me if I am mistaken but should part 2 not also be 5 DOF?
The massless spring between two particles is the analogy that is used to describe a diatomic molecule.
We have 2 translational DOF, 2 rotational DOF and vibrational DOF.

 

[ehild]

Correct me if I am mistaken but should part 2 not also be 5 DOF?
The massless spring between two particles is the analogy that is used to describe a diatomic molecule.
We have 2 translational DOF, 2 rotational DOF and vibrational DOF.

You use the formula for number of degrees of freedom = Nn - k where N is the number of particles and n is the number of dimensions. A plane has two dimensions. There are two particles. There are no constraints.
On the plane, a two-atomic molecule has only one kind of rotation: with axis, perpendicular to the plane. It can not move out of the plane. And it has one kind of vibration.

It is a different thing that applying equipartition principle, the vibrational degrees of freedom count twice when calculating the average energy of the molecule, as vibrational energy is the sum of KE and elastic potential energy.

ehild