Question

A smooth wire is bent into the shape of a helix, with cylindrical polar coordinates \(\rho=R\) and \(z=\lambda \phi,\) where \(R\) and \(\lambda\) are constants and the \(z\) axis is vertically up (and gravity vertically down). Using \(z\) as your generalized coordinate, write down the Lagrangian for a bead of mass \(m\) threaded on the wire. Find the Lagrange equation and hence the bead's vertical acceleration \(\ddot{z}\). In the limit that \(R \rightarrow 0\), what is \(\ddot{z} ?\) Does this make sense?

Short answer

In the limit \( R \rightarrow 0 \), \( \ddot{z} = -g \). This is consistent with free fall.

Step-by-step solution

Identify the Generalized Coordinate

The problem specifies using \( z \) as the generalized coordinate. Note that \( z = \lambda \phi \), meaning as the bead moves along the wire, \( z \) changes with \( \phi \). This will relate the radial and vertical motion.

Express the Kinetic Energy

The bead can move in both the radial and vertical directions. In cylindrical coordinates, the velocity components are \( \dot{\rho} = 0 \), \( \rho \dot{\phi} = R \dot{\phi} \), and \( \dot{z} \). The kinetic energy \( T \) is given by:\[T = \frac{1}{2} m \left((R \dot{\phi})^2 + (\dot{z})^2 \right)\]Now, since \( z = \lambda \phi \), we have \( \dot{z} = \lambda \dot{\phi} \). Substitute this into the kinetic energy expression:\[T = \frac{1}{2} m \left( R^2 \dot{\phi}^2 + (\lambda \dot{\phi})^2 \right) = \frac{1}{2} m \dot{\phi}^2 (R^2 + \lambda^2)\]Using \( \dot{\phi} = \frac{\dot{z}}{\lambda} \), the kinetic energy becomes \( T = \frac{m \dot{z}^2}{2 \lambda^2}(R^2 + \lambda^2) \).

Express the Potential Energy

The potential energy \( V \) is due to gravity, which is dependent on the vertical height. Therefore, \( V = mgz \).

Write the Lagrangian

The Lagrangian \( \mathcal{L} \) is defined as the difference between the kinetic and potential energies:\[\mathcal{L} = T - V = \frac{m}{2 \lambda^2} (R^2 + \lambda^2) \dot{z}^2 - mgz\]

Derive the Euler-Lagrange Equation

Using the Euler-Lagrange equation \( \frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{z}}\right) - \frac{\partial \mathcal{L}}{\partial z} = 0 \), find the expression:1. Calculate \( \frac{\partial \mathcal{L}}{\partial \dot{z}} = \frac{m}{\lambda^2}(R^2 + \lambda^2) \dot{z} \)2. Then compute \( \frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{z}}\right) = \frac{m}{\lambda^2}(R^2 + \lambda^2) \ddot{z} \)3. Finally, calculate \( \frac{\partial \mathcal{L}}{\partial z} = -mg \)Substitute into the Euler-Lagrange equation:\[\frac{m}{\lambda^2}(R^2 + \lambda^2) \ddot{z} + mg = 0\]

Solve for Vertical Acceleration \(\ddot{z}\)

From the Euler-Lagrange equation, isolate \( \ddot{z} \):\[\ddot{z} = -\frac{\lambda^2 g}{R^2 + \lambda^2}\]

Evaluate the Limit \( R \rightarrow 0 \)

Substitute \( R = 0 \) into the expression for \( \ddot{z} \):\[\ddot{z} = -g\]This matches the free-fall acceleration due to gravity, indicating that in this limit, the bead would fall freely as if there were no constraints (the wire becomes a vertical line).

Conclusion

When \( R \rightarrow 0 \), \( \ddot{z} = -g \) describes the bead's acceleration under gravity alone, with no radial constraint. This makes physical sense as the helix collapses into a vertical line.

Key concepts

Helix

A helix is a fascinating geometric shape that resembles a spiral or coil. Imagine a wire twisted in a perfect spring-like form. In our exercise, the helix is not just any helix; it has a constant radius and is defined in cylindrical polar coordinates. This type of helix has a fixed cylindrical radius, denoted as \( \rho = R \), which determines how far it is from the central axis at any point. The vertical position \( z \) relates to the angular displacement \( \phi \) through the equation \( z = \lambda \phi \). This means that for every complete circle around the helix, there is a fixed vertical rise \( \lambda \). This constant rise-to-turn ratio gives the helix its uniformly spiraling nature. Understanding these properties helps in solving problems involving motion along helices, where both radial and vertical motions are intricately linked.

Cylindrical Polar Coordinates

Cylindrical polar coordinates are a three-dimensional coordinate system useful for describing positions in space where rotational symmetry exists. In this system, any point in space is described using three values: the radial distance \( \rho \), the azimuthal angle \( \phi \), and the height \( z \). Try to envision it as a combination of polar coordinates, which you might use on a flat circle, with an added vertical component. This makes it particularly useful for problems involving round, cylindrical, or spiral shapes, such as the helix in our problem.

• \( \rho \) indicates how far from the cylinder's central axis the point is.
• \( \phi \) describes the angle around that axis, much like longitude on Earth.
• \( z \) is simply the vertical height, indicating how high or low a point is.

By understanding this coordinate system, you can better describe and analyze movement in spirals and rotations.

Vertical Acceleration

Vertical acceleration refers to the change in velocity along the vertical axis over time. In this exercise, we consider the motion of a bead along a helix, which involves both radial and vertical components. To find the bead's vertical acceleration, we employ Lagrangian mechanics, which simplifies the process of dealing with complex motion by focusing on energy rather than forces. Using the Lagrangian, which is derived from the difference between kinetic and potential energy, we apply the Euler-Lagrange equation to determine \( \ddot{z} \), the vertical acceleration.
The result \( \ddot{z} = -\frac{\lambda^2 g}{R^2 + \lambda^2} \) shows how both the geometry of the helix and gravitational force influence the bead’s acceleration. Notably, when \( R \rightarrow 0 \), the vertical acceleration simplifies to \( -g \), signifying free fall under gravity without radial constraint. This insight highlights the beauty of how theoretical physics matches our expectations based on everyday experiences.