Chapter 7: Problem 22
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equation of motion and show that it is the same as that for a normal, nonaccelerating pendulum, except that \(g\) has been replaced by \(g+a\).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 \(\sqrt{(g+a) / l}\).
Short Answer
Step by step solution
Understand the Physical Setup
Set Up the Coordinate System
Calculate the Bob's Velocity Components
Write Down the Kinetic Energy (T)
Write Down the Potential Energy (V)
Form the Lagrangian
Derive the Euler-Lagrange Equation
Find the Equation of Motion
Linearize for Small Oscillations
Identify Angular Frequency
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Simple Pendulum
This setup changes how we calculate forces acting on the pendulum. Typically, a simple pendulum's motion is influenced by gravity alone. However, here the upward acceleration of the elevator adds an extra force, similar to increasing gravity.
The angular displacement, which is how far the pendulum deviates from its resting vertical position, is represented by the angle \( \phi \). This angle is crucial for analyzing the pendulum's motion in terms of Lagrangian mechanics.
Euler-Lagrange Equation
The Lagrangian \( \mathcal{L} \) is expressed as \( \mathcal{L} = T - V\), where \( T \) is the kinetic energy and \( V \) is the potential energy. To find the pendulum's equation of motion, we apply the Euler-Lagrange equation: \[ \frac{d}{dt} \frac{\partial \mathcal{L}}{\partial \dot{\phi}} - \frac{\partial \mathcal{L}}{\partial \phi} = 0 \]
This equation yields a differential equation that governs how \( \phi \) changes over time, which is vital to understanding how the pendulum behaves dynamically.
Equations of Motion
Through the Euler-Lagrange approach described previously, we derived the equation \[ \ddot{\phi} + \frac{g+a}{l} \sin(\phi) = 0 \]
This second-order differential equation describes how the angular position \( \phi \) changes. For small oscillations, we simplify this by approximating \( \sin(\phi) \approx \phi \), resulting in a linear differential equation: \[ \ddot{\phi} + \frac{g+a}{l} \phi = 0 \]
This linearization is crucial as it allows us to delve into simple harmonic motion, a staple in classical mechanics, providing insights into how systems oscillate around a stable point.
Angular Frequency
From the simplified equation of motion, the angular frequency \( \omega \) is determined by the expression \[ \omega = \sqrt{\frac{g+a}{l}} \]
This formula highlights how the effective gravitational force—adjusted by both gravity and the elevator's acceleration—affects the rate of oscillation. Understanding angular frequency is valuable for predicting how fast the pendulum completes its back-and-forth swings, which can be directly correlated to periodic behavior seen throughout physics.
It also shows how adjustments in either gravitational forces or pendulum length \( l \) alter the pendulum's responsive timing to those forces.