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Chapter 4: Problem 1

(Validity of overdamped limit) Find the conditions under which it is valid to approximate the equation \(m L^{2} \vec{\theta}+b \dot{\theta}+m g L \sin \theta=\Gamma\) by its overdamped limit \(b \dot{\theta}+m g L \sin \theta=\Gamma\).

Short Answer

Expert verified
The overdamped limit of the equation \(m L^{2} \vec{\theta}+b \dot{\theta}+m g L \sin \theta=\Gamma\) is a valid approximation when the angular acceleration is much smaller than the ratio of the damping term to the moment of inertia: \[\vec{\theta} \ll \frac{b \dot{\theta}}{m L^2}\]

Step by step solution

01

In the given equation of motion, the term \(m L^{2} \vec{\theta}\) is removed in the overdamped limit. This term is responsible for the oscillatory motion of the pendulum, while the other two terms are responsible for the damping and the driving force, respectively. #Step 2: Examine the ratio of the removed term to the other terms#

To find the conditions under which it's valid to remove the term \(m L^{2} \vec{\theta}\), we can examine the ratio of this term to the other terms in the equation. Let's divide the entire equation by \(b \dot{\theta}\): \[\frac{m L^{2} \vec{\theta}}{b \dot{\theta}} + 1 + \frac{m g L \sin \theta}{b \dot{\theta}} = \frac{\Gamma}{b \dot{\theta}}\] The overdamped approximation implies that the term \(\frac{m L^{2} \vec{\theta}}{b \dot{\theta}}\) becomes negligible, which leads to: \[1 + \frac{m g L \sin \theta}{b \dot{\theta}} \approx \frac{\Gamma}{b \dot{\theta}}\] #Step 3: Find the condition for the overdamped limit#
02

Based on our analysis, the overdamped limit is a valid approximation when the ratio \(\frac{m L^{2} \vec{\theta}}{b \dot{\theta}}\) is much less than 1. This implies that \(m L^{2} \vec{\theta} \ll b \dot{\theta}\) Dividing both sides by \(m L^2\) gives \[\vec{\theta} \ll \frac{b \dot{\theta}}{m L^2}\] This condition tells us that the overdamped limit is valid when the angular acceleration is much smaller than the ratio of the damping term to the moment of inertia. #Step 4: Conclude with the validity condition#

In conclusion, the overdamped limit of the equation \(m L^{2} \vec{\theta}+b \dot{\theta}+m g L \sin \theta=\Gamma\) is a valid approximation when the angular acceleration is much smaller than the ratio of the damping term to the moment of inertia: \[\vec{\theta} \ll \frac{b \dot{\theta}}{m L^2}\]

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Key Concepts

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

Overdamped Approximation in Physics
The concept of an overdamped system is critical in understanding how oscillations behave under high damping. When a system is overdamped, the damping force is large enough that it prevents the system from oscillating freely. Instead of oscillating back and forth, an overdamped system returns to equilibrium without oscillating past its rest position. This behavior is common in systems where energy dissipation is high compared to the restoring forces.

In physics, making an overdamped approximation means assuming that the effects of the oscillatory motion are negligible, and the dominating factor is the damping force. In the context of the exercise provided, the overdamped limit leads to the removal of the term representing the oscillatory motion, \(m L^{2} \vec{\theta}\), leaving only the damping and the driving force terms in the equation. This simplification is valid under certain conditions where the system's response is dominated by viscous forces rather than inertia.
Damping in Oscillatory Systems
Oscillatory systems can have different types of damping, which drastically affect their behavior. Damping refers to the presence of a force that opposes the motion, usually a result of friction or resistance, which in turn dissipates energy from the system. The resistance can come from various sources like air resistance, internal friction within the material, or external forces opposing the movement.

An oscillatory system without any damping will continue to move indefinitely with a constant amplitude, a condition known as simple harmonic motion. The addition of damping causes the amplitude of the oscillations to decrease over time. Depending on the amount of damping, the system can be classified as underdamped, critically damped, or overdamped. It is the overdamped systems that markedly avoid repeating oscillations and instead creep back to a state of rest, as highlighted in the provided exercise.
Angular Acceleration
Angular acceleration is a measure of how quickly the angular velocity of a rotating object changes with time. It is the rotational equivalent to linear acceleration but for objects rotating around an axis or pivot. Defined as the rate of change of angular velocity, angular acceleration has the units of radians per second squared (\(rad/s^2\)).

In the equation of motion provided in the exercise, the term \(m L^{2} \vec{\theta}\) includes the angular acceleration, \(\vec{\theta}\), that describes how rapidly the angular velocity changes. When this change is significantly small compared to the damping force, the oscillatory component of the motion becomes negligible, leading to the overdamped approximation where the term related to angular acceleration is removed from the equation.
Moment of Inertia
The moment of inertia is a physical quantity that determines how difficult it is to change the rotational motion of an object. It is the rotational equivalent of mass for linear movements 鈥� just as mass resists linear acceleration, moment of inertia resists angular acceleration. In the formulation \(m L^{2}\), which appears in the pendulum equation from the exercise, this product represents the moment of inertia of the pendulum with respect to its pivot point.

The moment of inertia depends on how the mass is distributed with respect to the axis of rotation. Closer mass to the rotational axis means less inertia, making it easier to accelerate rotationally; mass further away increases the moment of inertia. As seen in the exercise analysis, when the moment of inertia is large relative to the damping term, \(b \dot{\theta}\), the angular acceleration is low enough that the overdamped approximation is suitable, simplifying the system's analysis.

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Most popular questions from this chapter

(Nongeneric scaling laws) In deriving the square-root scaling law for the time spent passing through a bottleneck, we assumed that \(\dot{x}\) had a quadratic minimum. This is the generic case, but what if the minimum were of higher order? Suppose that the bottleneck is governed by \(\dot{x}=r+x^{2 n}\), where \(n>1\) is an integer. Using the method of Exercise 4.3.9, show that \(T_{\text {twekreck }} \approx c r^{6}\), and determine \(b\) and \(c\). (It's acceptable to leave \(c\) in the form of a definite integral. If you know complex variables and residue theory, you should be able to evaluate \(c\) exactly by integrating around the boundary of the pie-slice \(\left\\{z=r e^{i \theta}: 0 \leq \theta \leq \pi / n, 0 \leq r \leq R\right\\}\) and letting \(R \rightarrow \infty\).)

(Torsional spring) Suppose that our overdamped pendulum is connected to a torsional spring. As the pendulum rotates, the spring winds up and gencrates an opposing torque \(-k \theta\). Then the equation of motion becomes \(b \theta+\) \(m g L \sin \theta=\Gamma-k \theta\) a) Does this equation give a well-defined vector field on the circle? b) Nondimensionalize the equation. c) What does the pendulum do in the long run? d) Show that many bifurcations occur as \(k\) is varied from 0 to \(\infty\). What kind of bifurcations are they?

The oscillation period for the nonuniform oscillator is given by the integral \(T=\int_{\pi}^{k} \frac{d \theta}{\omega-a \sin \theta}\), where \(\omega>a>0\). Evaluate this integral as follows. a) Let \(u=\tan \frac{\theta}{2} .\) Solve for \(\theta\) and then express \(d \theta\) in terms of \(u\) and \(d u\). b) Show that \(\sin \theta=2 u /\left(I+u^{2}\right)\). (Hint: Draw a right triangle with base 1 and height \(u .\) Then \(\frac{\theta}{2}\) is the angle opposite the side of length \(u\), since \(u=\tan \frac{g}{2}\) by definition. Finally, invoke the half-angle formula \(\sin \theta=2 \sin \frac{f}{2} \cos \frac{f}{2}\).) c) Show that \(u \rightarrow \pm \infty\) as \(\theta \rightarrow \pm \pi\), and use that fact to rewrite the limits of integration. d) Express \(T\) as an integral with respect to \(u\). e) Finally, complete the square in the denominator of the integrand of (d), and reduce the integral to the one studied in Exercise \(4.3 .1\), for a suitable choice of \(x\) and \(r\).

For each of the following vector fields, find and classify all the fixed points, and sketch the phase portrait on the circle. $$ \dot{\theta}=\sin k \theta \text { where } k \text { is a positive integer. } $$

(Current and voltage oscillations) Consider a Josephson junction in the overdamped limit \(\beta=0\) a) Sketch the supercurrent \(I_{e} \sin \phi(t)\) as a function of \(t\), assuming first that \(I / I_{e}\) is slightly greater than 1 , and then assuming that \(I / I_{c} \gg 1 .\) (Hint: In cach case, visualize the flow on the circle, as given by Equation \((4.6 .7)\).) b) Sketch the instantaneous voltage \(V(t)\) for the two cases considered in (a).
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