Bead moving down a Helical Wire subject to Constraints

In summary, the study analyzes the dynamics of a bead sliding along a helical wire under specific constraints. It explores the forces acting on the bead, including gravitational and tension forces, while considering the geometry of the helical wire. The motion of the bead is influenced by factors such as its mass, the angle of the helix, and the effects of friction. The findings contribute to understanding the interactions between constrained motion and physical forces in mechanical systems.
  • #1
deuteron
60
13
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##.
 
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  • #2
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?
 
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  • #3
I think h is the pitch.
 
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  • #4
Gordianus said:
I think h is the pitch.
I agree.

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

Cylindrical.png
 
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  • #5
Lnewqban said:
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?
 
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  • #6
erobz said:
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
 
  • #7
deuteron said:
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?
 
  • #8
erobz said:
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.)
 
  • #9
renormalize said:
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.
 
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  • #10
erobz said:
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.
 
  • #11
Lnewqban said:
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?
 
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  • #12
Sorry for the late reply, it is clear now! Thanks everyone!
 
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FAQ: Bead moving down a Helical Wire subject to Constraints

What is the basic setup of the problem involving a bead moving down a helical wire?

The problem involves a bead that is constrained to move along a helical wire. The helical wire can be described by its radius, pitch (the vertical distance between turns), and the number of turns. The bead is influenced by gravitational force and possibly other forces like friction, but it is constrained to stay on the helical path defined by the wire.

How do you mathematically describe the helical path of the wire?

The helical path can be parameterized using cylindrical coordinates. If we let the radius of the helix be \( R \) and the pitch be \( P \), the position of the bead as a function of the parameter \( t \) (which could represent time or an angular parameter) can be given by \( x(t) = R \cos(t) \), \( y(t) = R \sin(t) \), and \( z(t) = \frac{P}{2\pi} t \). Here, \( t \) represents the angle in radians.

What are the constraints acting on the bead?

The primary constraint is that the bead must remain on the helical wire. This implies that its motion is restricted to the path defined by the helix. Mathematically, this can be expressed as a constraint equation that the bead's position must satisfy at all times. Additionally, if there is friction, it can be considered as a constraint force that acts tangentially along the wire.

How do you derive the equations of motion for the bead?

The equations of motion can be derived using the Lagrangian mechanics framework. The Lagrangian \( L \) is defined as the difference between the kinetic energy \( T \) and the potential energy \( V \) of the system. For a bead of mass \( m \) on a helical wire with gravitational potential energy, \( V = mgz \), the kinetic energy \( T \) can be expressed in terms of the bead's velocity components. The constraint equations are used to eliminate dependent coordinates, and the Euler-Lagrange equations are applied to obtain the equations of motion.

How does gravity affect the motion of the bead on the helical wire?

Gravity acts downward along the z-axis and influences the bead's motion by providing a constant force in the negative z-direction. This force contributes to the potential energy of the system and results in a component of acceleration along the helix. The gravitational force causes the bead to accelerate down the helical path, and the specific trajectory will depend on the balance between gravitational force, the normal force from the wire, and any frictional forces present.

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