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Chapter 7: Problem 20

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

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

Step by step solution

01

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.
02

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 \( T = \frac{m \dot{z}^2}{2 \lambda^2}(R^2 + \lambda^2) \).
03

Express the Potential Energy

The potential energy \( V \) is due to gravity, which is dependent on the vertical height. Therefore, \( V = mgz \).
04

Write the Lagrangian

The Lagrangian \( \mathcal{L} \) is defined as the difference between the kinetic and potential energies:\[\mathcal{L} = T - V = \frac{m}{2 \lambda^2} (R^2 + \lambda^2) \dot{z}^2 - mgz\]
05

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:\[\frac{m}{\lambda^2}(R^2 + \lambda^2) \ddot{z} + mg = 0\]
06

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

From the Euler-Lagrange equation, isolate \( \ddot{z} \):\[\ddot{z} = -\frac{\lambda^2 g}{R^2 + \lambda^2}\]
07

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).
08

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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.

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Most popular questions from this chapter

Using the usual angle \(\phi\) as generalized coordinate, write down the Lagrangian for a simple pendulum of length \(l\) suspended from the ceiling of an elevator that is accelerating upward with constant acceleration \(a\). (Be careful when writing \(T\); it is probably safest to write the bob's velocity in component form.) Find the Lagrange equation of motion and show that it is the same as that for a normal, nonaccelerating pendulum, except that \(g\) has been replaced by \(g+a\). In particular, the angular frequency of small oscillations is \(\sqrt{(g+a) / l}\).

A particle is confined to move on the surface of a circular cone with its axis on the \(z\) axis, vertex at the origin (pointing down), and half-angle \(\alpha\). The particle's position can be specified by two generalized coordinates, which you can choose to be the coordinates \((\rho, \phi)\) of cylindrical polar coordinates. Write down the equations that give the three Cartesian coordinates of the particle in terms of the generalized coordinates ( \(\rho, \phi\) ) and vice versa.

Find the components of \(\nabla f(r, \phi)\) in two-dimensional polar coordinates. [Hint: Remember that the change in the scalar \(f \text { as a result of an infinitesimal displacement } d \mathbf{r} \text { is } d f=\nabla f \cdot d \mathbf{r}.]\)

A mass \(m_{1}\) rests on a frictionless horizontal table and is attached to a massless string. The string runs horizontally to the edge of the table, where it passes over a massless, frictionless pulley and then hangs vertically down. A second mass \(m_{2}\) is now attached to the bottom end of the string. Write down the Lagrangian for the system. Find the Lagrange equation of motion, and solve it for the acceleration of the blocks. For your generalized coordinate, use the distance \(x\) of the second mass below the tabletop.

Consider two particles moving unconstrained in three dimensions, with potential energy \(U\left(\mathbf{r}_{1}, \mathbf{r}_{2}\right) .\) (a) Write down the six equations of motion obtained by applying Newton's second law to each particle. (b) Write down the Lagrangian \(\mathcal{L}\left(\mathbf{r}_{1}, \mathbf{r}_{2}, \dot{\mathbf{r}}_{1}, \dot{\mathbf{r}}_{2}\right)=T-U\) and show that the six Lagrange equations are the same as the six Newtonian equations of part (a). This establishes the validity of Lagrange's equations in rectangular coordinates, which in turn establishes Hamilton's principle. since the latter is independent of coordinates, this proves Lagrange's equations in any coordinate system.
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