Bead moving down a Helical Wire subject to Constraints

[deuteron]

Homework Statement

What is the constraint for the bead on a helix wire moving under gravitation ignoring friction?

Relevant Equations

##q=\{r,\phi,z\}\ \hat=## cylindrical coordinates

One of the constraints is given as ##r=R##, which is very obvious. The second constraint is however given as

$$\phi - \frac {2\pi} h z=0$$

where ##h## is the increase of ##z## in one turn of the helix. Physically, I can't see where this constraint comes from and how ##\phi=\frac {2\pi}h z##.

 

[erobz]

I think ##h## is the total height of the helix, since it has a constant slope, ##\phi## is the angle turned as a function of the vertical position ##z##

Is there a digram of the helix that would contradict that?

 

[Gordianus]

I think h is the pitch.

 

[Lnewqban]

I think h is the pitch.

I agree.

@deuteron
Please, see:
https://en.wikipedia.org/wiki/Cylindrical_coordinate_system

 

[erobz]

I agree.

Can you explain the ##2 \pi## in the numerator? The pitch is the vertical rise per unit angle turned. So lets say the pitch is ## h = \frac{1 \text{[m]}}{ 2 \pi \text{[rad]}}##, if we let ##z## be ##1 \text{[m]} ##, then the angle turned ##\phi## would be ## 4 \pi^2 \text{[rad]} ## according to the formula...that seems to be a contradiction?

 

[deuteron]

I think ##h## is the total height of the helix, since it has a constant slope, ##\phi## is the angle turned as a function of the vertical position ##z##

Is there a digram of the helix that would contradict that?

There isn't a diagram but I edited the question to clarify what ##h## is, it is given as the increase of ##z## in one turn

 

[erobz]

There isn't a diagram but I edited the question to clarify what ##h## is, it is given as the increase of ##z## in one turn

So if ##h## is indeed the pitch, am I having a brain fart in post #5?

 

[renormalize]

Can you explain the ##2 \pi## in the numerator? The pitch is the vertical rise per unit angle turned. So lets say the pitch is ## h = \frac{1 \text{[m]}}{ 2 \pi \text{[rad]}}##, if we let ##z## be ##1 \text{[m]} ##, then the angle turned ##\phi## would be ## 4 \pi^2 \text{[rad]} ## according to the formula...that seems to be a contradiction?

Wrong definition of pitch. From Wikipedia (https://en.wikipedia.org/wiki/Helix):
"The pitch of a helix is the height of one complete helix turn, measured parallel to the axis of the helix." (Emphasis added.)

 

[erobz]

Wrong definition of pitch. From Wikipedia (https://en.wikipedia.org/wiki/Helix):
"The pitch of a helix is the height of one complete helix turn, measured parallel to the axis of the helix." (Emphasis added.)

I guess I should have checked the definition. Thanks. @deuteron sorry for any confusion.

 

[Lnewqban]

Can you explain the ##2 \pi## in the numerator?

Hi @erobz
Sorry about delayed answer.
Is this still confusing?
I agreed because I believed that the values of h and z should be equal for one full turn (2π radians) or rotation of the particle.

 

[erobz]

Hi @erobz
Sorry about delayed answer.
Is this still confusing?
I agreed because I believed that the values of h and z should be equal for one full turn (2π radians) or rotation of the particle.

@renormalize set me straight. I assumed an incorrect definition of pitch for a helix. I don't know if its still confusing for the OP @deuteron however?

 

[deuteron]

Sorry for the late reply, it is clear now! Thanks everyone!