Question

Consider again the pendulum equations (sce Problem \(21)\) $$ d x / d t=y, \quad d y / d t=-4 \sin x $$ If the pendulum is set in motion from its downward equilibrium position with angular velocity \(v,\) then the initial conditions are \(x(0)=0, y(0)=v\) (a) Plot \(x\) versus \(t\) for \(v=2\) and also for \(v=5 .\) Explain the differing motions of the pendulum that these two graphs represent. (b) There is a critical value of \(v,\) which we denote by \(v_{c}\), such that one type of motion occurs for \(vv_{c}\). Estimate the value of \(v_{c} .\)

Short answer

The purpose of estimating the critical angular velocity (\(v_c\)) is to identify a value of \(v\) that separates the two types of motion, so the pendulum motion changes significantly for \(v>v_c\) compared to \(v<v_c\). This will help in understanding how the behavior of the pendulum's motion changes as the initial angular velocity increases.

Step-by-step solution

Set up the system of differential equations

We are given the following system of differential equations: $$ \frac{dx}{dt} = y, \quad \frac{dy}{dt} = -4\sin x $$ With the initial conditions: $$ x(0) = 0, \quad y(0) = v $$ We will solve the system of equations for different values of the initial angular velocity \(v\) and analyze the resulting motion of the pendulum.

Solve for \(x(t)\) and \(y(t)\) with \(v = 2\)

We use a numerical method, such as the Euler or Runge-Kutta method, to solve the system of differential equations with \(v = 2\): $$ x(0) = 0, \quad y(0) = 2 $$ We'll then obtain the solution for \(x(t)\) and plot it as a function of time.

Solve for \(x(t)\) and \(y(t)\) with \(v = 5\)

Now, we use the same numerical method to solve the system of differential equations with \(v = 5\): $$ x(0) = 0, \quad y(0) = 5 $$ We'll obtain the solution for \(x(t)\) and plot it as a function of time.

Compare the graphs

Once we have the graphs for \(x(t)\) with \(v = 2\) and \(v = 5\), we analyze their shapes to determine the differing motions of the pendulum they represent. For example, we can observe the magnitude of oscillations, their periods, and whether the pendulum passes through its vertical equilibrium.

Estimate the critical angular velocity \(v_c\)

To find the critical angular velocity \(v_c\), we'll iteratively solve the system of differential equations with different values of \(v\) and analyze the solutions' graphical behavior. The goal is to identify a value of \(v\) that separates the two types of motion, so the pendulum motion changes significantly for \(v>v_c\) compared to \(v<v_c\). We can use repeated trial-and-error or implement a more systematic approach, such as a bisection method, to estimate \(v_c\).

Key concepts

Differential Equations

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are fundamental in expressing the laws that govern the physical world, from simple mechanics to complex dynamical systems like weather patterns.

In the case of a pendulum, the equations are a set of two coupled first-order ordinary differential equations (ODEs) that describe its motion over time. The first equation, \(\frac{dx}{dt} = y\), relates to the velocity of the pendulum, while the second, \(\frac{dy}{dt} = -4\sin x\), derives from Newton's second law of motion and represents the acceleration, factoring in the gravitational force and the angle of displacement \(x\).

The complexity of such equations means they often cannot be solved using simple algebraic methods, thus requiring more advanced techniques to find their solutions.

Numerical Methods

Numerical methods are techniques used to approximate solutions for mathematical problems that are difficult or impossible to solve analytically. These methods convert continuous problems, such as differential equations, into discretized forms that can be computed using iteration.

By breaking down the continuous time into small intervals, numerical methods incrementally build the solution step by step. For the pendulum exercise, numerical methods allow us to compute the values of \(x(t)\) and \(y(t)\), which represent the angle and angular velocity of the pendulum, using the initial conditions provided.

Numerical methods are crucial in fields such as engineering, physics, and finance, where precise models of the real world are necessary. These methods include but are not limited to the Euler method, Trapezoidal rule, and the Runge-Kutta methods.

Initial Value Problem

An initial value problem is a type of differential equation along with a specific set of starting conditions, known as initial values. For a successful simulation of any physical system, you need both the rules (the differential equations) and the starting point (the initial conditions).

In our pendulum example, the initial value problem is defined by the pendulum's starting angle \(x(0) = 0\) and its initial angular velocity \(y(0) = v\). These conditions provide a snapshot of the system at \(t = 0\), allowing the numerical method to begin its iterative process. Without this information, the future state of the system would remain undetermined, as there can be infinitely many solutions to a differential equation without initial values.

Critical Angular Velocity

Critical angular velocity, denoted as \(v_c\), is an important concept in dynamics, especially in the study of oscillatory systems like pendulums. It is the threshold velocity at which the behavior of the pendulum's motion qualitatively changes between different states, such as moving from simple oscillation to a complete circular motion.

In the pendulum exercise, calculating \(v_c\) helps us understand the transition point where the pendulum goes from back-and-forth oscillations to spinning all the way around its pivot. This transition is a result of the pendulum receiving enough energy to overcome the gravitational force acting against it at the top of its arc. Estimating \(v_c\) is crucial, for instance, in designing clocks or amusement park rides, where the safety and functionality of the mechanism could depend on the pendulum activity remaining in a particular state of motion.

Runge-Kutta Method

The Runge-Kutta method is a powerful numerical algorithm used to solve ordinary differential equations, particularly useful when dealing with more complicated or stiff equations. It's essentially an advanced version of the Euler method, providing greater stability and accuracy.

The most commonly used Runge-Kutta method is the fourth-order method, often abbreviated as RK4, which calculates the system's state at incremental time steps. RK4 takes into account not just the initial slope (as with Euler's method), but also the slopes at the midpoint and end of the interval to determine the next value. This accounts for changes in the system's behavior between steps, resulting in a more accurate solution.

For the pendulum's motion, applying the Runge-Kutta method gives us a precise approximation of the angle and angular velocity over time, which is essential for understanding complex dynamical behavior like chaos or non-linear oscillation.