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
Step by step solution
Identify the Generalized Coordinate
Express the Kinetic Energy
Express the Potential Energy
Write the Lagrangian
Derive the Euler-Lagrange Equation
Solve for Vertical Acceleration \(\ddot{z}\)
Evaluate the Limit \( R \rightarrow 0 \)
Conclusion
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Helix
Cylindrical Polar Coordinates
- \( \rho \) 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.
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.