Question

The motion of a certain undamped pendulum is described by the equations $$ d x / d t=y, \quad d y / d t=-4 \sin x $$ If the pendulum is set in motion with an angular displacement \(A\) and no initial velocity, then the initial conditions are \(x(0)=A, y(0)=0\) (a) Let \(A=0.25\) and plot \(x\) versus \(t\). From the graph estimate the amplitude \(R\) and period \(T\) of the resulting motion of the pendulum. (b) Repeat part (a) for \(A=0.5,1.0,1.5,\) and \(2.0 .\) (c) How do the amplitude and period of the pendulum's motion depend on the initial position \(A^{7}\) Draw a graph to show each of these relationships. Can you say anything about the limiting value of the period as \(A \rightarrow 0 ?\) (d) Let \(A=4\) and plot \(x\) versus \(t\) Explain why this graph differs from those in parts (a) and (b). For what value of \(A\) does the transition take place?

Short answer

Short Answer: To analyze the relationship between the amplitude (A) and period (T) of a pendulum's motion, given its differential equations and initial conditions, integrate the differential equations and plot x against t for varying values of A. Evaluate the resulting graphs to estimate the amplitude (R) and period (T) and determine their relationships with A. Upon analyzing these relationships, the graph will reveal any patterns or trends. Additionally, consider the limiting value of T as A approaches 0 and compare the graphs when A equals different values. Finally, determine when a transition takes place, and provide explanations for any observed differences.

Step-by-step solution

Set up and solve the system of differential equations for a given value of A

To solve this problem, we need to integrate the given system of differential equations. There are various numerical techniques to do this, like the Euler method or the method of Runge-Kutta. You can also use specialized software or tools like MATLAB, Python, or Mathematica to solve the equations. Here, we'll use a general purpose numerical solver to obtain the solution to the system of differential equations. The initial conditions will be \(x(0)=A\) and \(y(0)=0\).

Plot x versus t for given values of A

After solving the system of differential equations for a given value of A, you should plot the graph of the angular displacement, x, against time, t. Use the following values of A: 0.25, 0.5, 1.0, 1.5, and 2.0. Take note of the amplitude for each value A, which is the maximum displacement from resting position.

Estimate the amplitude R and period T

Based on the graphs obtained in step 2, estimate the amplitude R and the period T of the motion for each value of A. The amplitude R should be the maximum value of the displacement, while the period T is the time it takes for the pendulum to make one full oscillation (go from one maximum value to the next)

Analyze the relationship between initial position A and amplitude R

Analyze how amplitude R changes with varying initial position A, and plot the graph of R versus A. Note any patterns or trends in the data.

Analyze the relationship between initial position A and period T

Analyze how period T changes with varying initial position A, and plot the graph of T versus A. Note any patterns or trends in the data. You can also consider the limiting value of the period T as A approaches 0.

Plot x versus t for A=4 and analyze the graph

For A=4, solve the system of differential equations, plot the graph of x versus t, and compare the graph with previous ones obtained in step 2. Describe any differences in the graph and provide explanations for these differences. Determine the value of A when this transition takes place.

Key concepts

Differential Equations

Pendulum motion can be described using differential equations, which are mathematical equations involving derivatives that express how variables change over time. In our exercise, the system of equations

\[d \frac{x}{dt} = y, \quad d \frac{y}{dt} = -4 \sin x\]represents this motion. Here, `x` is the angular displacement, and `y` is the angular velocity of the pendulum. The equations say that the rate of change of displacement is the velocity, and the rate of change of velocity is proportional to the sine of the angle.

This system is nonlinear due to the sine function, which makes it more complex to solve analytically. Therefore, numerical methods are often employed to find approximate solutions over time so that we can understand how the pendulum behaves under varying initial conditions.

Initial Conditions

The initial conditions are critical in determining the unique solution to the differential equations. For a pendulum, they specify the initial displacement and velocity. In this exercise:

• \( x(0) = A \): The initial angular displacement.
• \( y(0) = 0 \): The pendulum starts at rest without initial velocity.

These conditions allow us to set a starting point for our analysis and simulation. If we set a different initial angular displacement `A`, it affects how far the pendulum can swing and how long it takes to complete each oscillation. Understanding various `A` values helps us see the pendulum's dynamic behavior, from gentle swings to more robust motions.

Numerical Methods

Since the differential equations for pendulum motion can be difficult to solve analytically, numerical methods come into play. These methods, such as the Euler method or the Runge-Kutta method, allow us to approximate solutions by breaking down the problem into smaller, manageable steps.

In practice, software and tools like MATLAB, Python, or Mathematica are utilized to perform these computations efficiently. These tools numerically integrate the equations over time, respecting the initial conditions provided. Numerical methods give us a detailed view of how `x` (displacement) changes with time `t`, allowing us to visualize the swinging motion of the pendulum under different initial conditions.

Amplitude and Period Analysis

By analyzing graphs of angular displacement versus time, we can estimate two important characteristics of pendulum motion: amplitude and period.

• Amplitude (R): The maximum angular displacement from the rest position. It shows how far the pendulum swings.
• Period (T): The time it takes for the pendulum to complete one full cycle (from maximum displacement back to the same point).

For small angles, the period is typically consistent, but as the initial displacement increases, nonlinear characteristics of the sine function affect the period. An interesting phenomenon occurs as the initial displacement `A` grows large or approaches zero, affecting the period. Plotting these characteristics helps visualize these functional dependencies and study how they evolve as `A` varies.