Good coordinates and degrees of freedom

[Sahar Ali]

I have to present a topic "Good coordinates and degree of freedom" I know what are good coordinate and degree of freedom. but I will have to explain examples/question given below(from Liboff's text) I know the answer to all of them but I really do not know how to explain these how will I explain these parts to a class while presenting anyone who can give a little explanation?
For each of the following systems, specify the number of degrees of freedom and a set of good coordinates.

(a) A bead constrained to move on a closed circular loop.
(b) A bean constrained to move on the helix of constant pitch and constant radius.
(c) A particle on a right circular cylinder.
(d) A pair of scissors on a plane.
(e) A rigid rod in 3-space.
(f) A rigid cross in 3-space.
(g) A linear spring in 3-space.
(h) Any rigid body with one point fixed.
(i) A Hydrogen atom
(j) A lithium atom
(k) A compound pendulum (two pendulums attached end to end)

My answers are:
(a) The distance along loop from an arbitrary fixed point on the loop. 1 degree of freedom.
(b) The distance along helix from an arbitrary fixed point on the helix. 1 degree of freedom.
(c) Cylindrical coordinates. 2 degrees of freedom.
(d) 3 numbers to locate the center of scissors. One for angle scissors makes with the chosen axis. One for angle scissors is open. 5 degrees of freedom.
(e) 3 numbers to locate the center of a rod in space. Two numbers to orient rod in space, typically q and f.5 degrees of freedom.
(f) 3 numbers to locate a center of the rod in space. Two numbers to orient the rod in space. Two numbers to rotate about both axes in space. 6 degrees of freedom.
(g) Three numbers to locate the center of spring in space, two numbers to orient spring in space and one number for amount spring is stretched. 5 degrees of freedom.
(h) 3 numbers to locate a body in space. 2 numbers to orient body and 2 numbers about each axis of rotation. 7 degrees of freedom
(i) 3 numbers to locate proton in space. 3 numbers to locate the electron in space. 6 degrees of freedom.
(j) 3 numbers to locate the nucleus in space. 3 numbers for each electron in space. 12 degrees of freedom.
(k) 2 degrees of freedom for the first pendulum. 2 degrees of freedom for the second pendulum.

 

[Orodruin]

(d) 3 numbers to locate the center of scissors. One for angle scissors makes with the chosen axis. One for angle scissors is open. 5 degrees of freedom.

The scissors are moving in a plane. Not in three dimensions.

(f) 3 numbers to locate a center of the rod in space. Two numbers to orient the rod in space. Two numbers to rotate about both axes in space. 6 degrees of freedom

6 is correct, but it is not equal to 3+2+2.

g) Three numbers to locate the center of spring in space, two numbers to orient spring in space and one number for amount spring is stretched. 5 degrees of freedom.

3+2+1=?

(h) 3 numbers to locate a body in space. 2 numbers to orient body and 2 numbers about each axis of rotation. 7 degrees of freedom

The body is fixed in one point. Your counting of rotation angles is also wrong.

(j) 3 numbers to locate the nucleus in space. 3 numbers for each electron in space. 12 degrees of freedom

This depends on how you consider the degrees of freedom of the nucleus. In general, it would be inconsistent to give all electrons full freedom and not the nucleons. You should specify that you disregard degrees of freedom internal to the nucleus.

(k) 2 degrees of freedom for the first pendulum. 2 degrees of freedom for the second pendulum

Assuming the penduli move completely independent, yes.

 

[Sahar Ali]

Cant, we consider spin motion in all these cases? what if we take both spin and linear motion?