Question

The motion of a pendulum in a resisting medium is governed by $$ \ddot{\theta}+2 k \dot{\theta}+q \sin \theta=0, \quad k>0, \quad q>0 . $$ Describe the nature of the critical points and sketch the trajectories. Let \(\theta=\phi+\pi .\) Then the preceding equation becomes $$ \ddot{\phi}+2 k \dot{\phi}-q \phi=0 $$ Make a detailed analysis.

Short answer

Based on the given differential equation \(\ddot{\theta}+2 k \dot{\theta}+q \sin \theta=0\), we found that the critical points occur when \(\sin\theta = 0\), specifically at \(\theta = n\pi, \quad n=0, \pm1, \pm2,\ldots\). The trajectories of the original equation are oscillatory with decreasing amplitude due to the damping term. After applying the substitution \(\theta=\phi+\pi\), we obtained the transformed equation \(\ddot{\phi}+2k\dot{\phi}+q\sin\phi=0\). This equation behaves similarly to the original equation; the critical points occur when \(\sin\phi=0\) at \(\phi=n\pi, \quad n=0, \pm1, \pm2,\ldots\), and the trajectories are oscillatory with their amplitude decreasing over time due to the dissipative term.

Step-by-step solution

Identify the critical points of the original equation

We have the differential equation: $$ \ddot{\theta}+2 k \dot{\theta}+q \sin \theta=0, \quad k>0, \quad q>0 $$ To find the critical points, we need to find the points where \(\dot{\theta}=0\) and \(\ddot{\theta}=0\). Setting \(\dot{\theta}=0\) and \(\ddot{\theta}=0\) we get: $$ 2k\dot{\theta}=0 \quad \text{and} \quad q\sin\theta=0 $$ Since \(k>0\), we must have \(\dot{\theta}=0\), and the critical points occur when \(q\sin\theta=0\). Since \(q>0\), this happens when \(\sin\theta = 0\), which means \(\theta = n\pi, \quad n=0, \pm1, \pm2,\ldots\)

Sketch the trajectories of the original equation

With the critical points identified, we can sketch the trajectories. Due to the damping term \(2k\dot{\theta}\), we know that the pendulum will lose energy over time and eventually come to rest. This means that the pendulum will typically oscillate between two critical points before settling at a critical point as the energy in the system dissipates. This is easier to draw than to explain, but qualitatively, the trajectories will have a decreasing amplitude and will approach the critical points as time goes on. Keep in mind that the pendulum can start at any initial angle with any initial velocity depending on the initial conditions.

Substitute the given substitution and simplify

We are given the substitution \(\theta=\phi+\pi\). Substituting this into the original equation, we get: $$ \ddot{\phi}+2k\dot{\phi}-q\sin(\phi+\pi)=0 $$ Using the trigonometric identity \(\sin(\alpha+\beta)=\sin\alpha\cos\beta+\cos\alpha\sin\beta\), we have: $$ \sin(\phi+\pi)=\sin\phi\cos\pi+\cos\phi\sin\pi=-\sin\phi $$ Thus, the equation becomes: $$ \ddot{\phi}+2k\dot{\phi}-q(-\sin\phi)=0 $$ Or, equivalently: $$ \ddot{\phi}+2k\dot{\phi}+q\sin\phi=0 $$

Analyze the transformed equation

Now, we need to analyze this transformed equation. Notice that the transformed equation is identical to the original equation, with the only difference being the change in the dependent variable from \(\theta\) to \(\phi\). This implies that the critical points, trajectories, and behavior of this transformed equation will be the same as the original equation. Therefore, we can infer the same conclusions about the critical points and trajectories as done in Step 1 and Step 2 with the substitution in place. The critical points occur when \(\sin\phi=0\), which happens at \(\phi=n\pi, \quad n=0, \pm1, \pm2,\ldots\). The trajectories will again be oscillatory with the amplitude decreasing over time due to the dissipative term, \(2k\dot{\phi}\).

Key concepts

Critical Points

Understanding critical points in the context of differential equations is essential for analyzing the behavior of dynamic systems, such as a pendulum in a resisting medium. For the differential equation \[\ddot{\theta}+2 k \dot{\theta}+q \sin \theta=0,\]a critical point is defined as a state in which the velocity \(\dot{\theta}=0\) and acceleration \(\ddot{\theta}=0\), indicating no motion. In this system, critical points occur when \(\sin\theta=0\), leading to positions at integer multiples of \(\pi\).

• These points represent the pendulum's equilibrium states, where it's at rest.
• Since we're dealing with a second-order differential equation, each critical point corresponds not just to a position, but also to a velocity - both being zero in this case.

Critical points are pivotal for determining the long-term behavior of systems and often serve as starting points for further analysis, such as stability and possible trajectories of motion.

Damping and Oscillation

The concept of damping and oscillation is integral to the study of mechanical systems and their motion equations. The given pendulum equation includes a damping coefficient \(2k\), representing the resistance of the medium through which the pendulum swings. In such a system:

• Damping is the force that continuously opposes the motion, causing the system to lose energy and gradually come to rest.
• Oscillation refers to the repetitive back and forth motion of the pendulum, which is influenced by the restoring force proportional to \(\sin\theta\).

The presence of damping ensures the oscillations will diminish over time, leading to a stabilized pendulum at one of its critical points. This reduction in amplitude is a hallmark of damped oscillatory systems and is a significant factor in predicting long-term behavior.

Pendulum Motion

The motion of a pendulum can be complex, but the essence of pendulum motion can be gleaned from its differential equation representation. In our case:\[\ddot{\theta}+2 k \dot{\theta}+q \sin \theta=0,\]where \(k\) is the damping coefficient and \(q\) is proportional to the gravitational force acting on the pendulum. Understanding pendulum motion involves recognizing how the components of this equation influence the behavior:

• The term \(\ddot{\theta}\) represents the angular acceleration or the rate of change of angular velocity.
• The term \(2k\dot{\theta}\) accounts for the damping, which acts to slow down the motion over time.
• The term \(q\sin\theta\) embodies the restoring force that pulls the pendulum back to its lowermost position, creating oscillation.

The resulting motion is thus a harmonious interplay between the restoring and damping forces, leading to oscillatory behavior that gradually ceases over time.

Trajectory Sketching

When it comes to trajectory sketching in the context of pendulum motion within a resisting medium, it's a graphical representation of how the pendulum's angle changes with time. This sketch will exhibit certain key characteristics:

• The amplitude of oscillation decreases over time due to the damping effect represented by the term \(2k\dot{\theta}\).
• Each oscillation will swing between two critical points until the pendulum comes to rest at a critical point.
• The shape of the trajectory depends on the initial conditions, such as initial angle and velocity.

Trajectory sketching is a useful tool for visualizing the behavior of dynamic systems, particularly how they evolve as they undergo oscillation and damping. Although we may not provide an exact trajectory here, understanding these qualitative features helps predict and analyze the system's behavior.