Differential Equations Day 19 -- Project 5 -- Pendular Motion

Directions

Follow the steps below and do your work in the provided GeoGebra windows. NOTE to VSC students taking this class with me: be sure you are accessing this through a GeoGebra Classroom Link! The URL of this page should have the word “classroom” in it. If not, then go back to Canvas and be sure to access this page from the Project 5 assignment. Also: I strongly recommend you login to GeoGebra.org with a free account so your work is saved, and you can come back later to review or modify it. For external readers: these "project" activities are meant to be taken as part of my course, so these sections of the GeoGebra book my not be as intelligible as others. My apologies.

Learning Goals

The primary goal of this project is to learn about Pendular Motion, a classical application of non linear second order homogeneous differential equations. A secondary goal is that you will practice using RK4 to estimate solutions of a second order differential equation.

Content Overview

When:

an object of mass
is suspended by a rod of length ; and
the rod is firmly attached to an anchor; and
damping due to immersion in a viscous fluid exists and is proportional to velocity of the object with proportionality coefficient ; and
this damping is also inversely proportional to the mass of the object

then, the angular displacement,

, of the object away from equilibrium (i.e. a "plumb line") is well modeled by a second order non linear differential equation:

Such systems are called simple pendulums. You can read more about the theoretical underpinning of this second order non linear differential equation model of simple pendulums here. Note that

is the acceleration due to gravity. Explore the applet below which illustrates the angular displacement

of such an object. You can explore by adjusting 4 parameters:

, the damping coefficient (a measurement of the viscosity of the fluid the system is immersed in)
, the length of the rod
, the mass of the object (in kilograms)
, the initial displacement of the pendulum from equilibrium (in radians)

Part 1 Steps

In the applet below for Part 1, set the mass to 9, the damping coefficient to 1, and the length to 7, and the initial angular displacement to 1.
Trace the angular displacement of object for the full time period (20).
Increase the damping coefficient to 2. Hold the other settings constant. Reset the time to 0. DO NOT CLEAR THE TRACE.
Trace the angular displacement of the object for the full time period.
Continue doubling the damping coefficient until damping appears to be over damped (i.e. angular displacement approaches plumb asymptotically).
Work down from your first damping coefficient that leads to over damping to find the approximate value of that leads critical damping (i.e. the smallest value of that leads to angular displacement that approaches plumb asymptotically)
Summarize your findings of the damping in the text box below the applet for Part 1. Be sure your written response uses complete sentences and is concise.

Part 1 Applet

Part 1 Text Box

Put your written response here.

Part 2 Steps

Consider the second order non linear differential equation

Use the method of systemification to transform the second order differential equation in into a system of first order differential equations in and .
As we've done in previous lessons, swap out and with and so that the system can be represented in the GeoGebra RK4 calculator below.
Update and below as appropriate to match your system.
Update the initial conditions below as appropriate.
Identify the RK4 estimate for with and type it in the text box below.

Part 2 Applet

Part 2 Text Box

Put your estimate of using here.

Part 3 Text Box

Use the applet at the start of this page (before Part 1) to experiment and attempt to answer the question: If and are held constant, what is the impact of increasing the mass ? Type your answer below. Be sure your written response uses complete sentences and is concise.

Part 4 (Challenge and Optional)

Use the applet at the start of this page (before Part 1) to experiment and attempt to answer the question: What value of is needed to ensure that a system of mass , and length is critically damped? In other words, attempt to the write a critical damped value for as a response to and . You can write as gamma.