CS426 Lecture 18 Modelling motion using ordinary differential equations (ode’s)

Flashcards for reviewing concepts related to modeling motion using ordinary differential equations.

What is Ordinary Differential Equation?

Contains derivative of one variable only

Give examples of physical systems that can be modeled by ODEs

SHM, Radioactive Decay, Linear Motion

What is the equation for SHM, and what do the variables stand for?

x = Asin (2ft), where x is displacement, A is amplitude, f is frequency, and t is time

What is the equation for Radioactive Decay, and what do the variables stand for?

N = N0e^(-t), where N is the number of atoms, N0 is the initial number of atoms,  is the decay constant, and t is time

What is the equation for Linear Motion, and what do the variables stand for?

x = ut + (1/2)at^2, where x is displacement, u is initial velocity, a is acceleration, and t is time

What is the equation for radioactive decay?

dN/dt = -N, N = N0e^(-t)

What is the half-life formula?

 = 0.693/t1/2

What is Euler's Method?

A method of numerical approximation to solve first order ODE

In the context of falling ball, what forces are considered?

Gravity, Drag, Inertia

Write the equations relating gravity,drag, and inertia

Fgravity=mg, Fdrag=-kv, FInertia=ma

How is the motion of the ball described?

By differential equation

What observation is used to rewrite the equation?

Velocity is change in position divided by change in time v=dy/dt

What are the two first order equations after rewriting the second order equation?

dy = vdt, dv = (g - (k/m)v)dt

What is required to advance balls position and velocity over time interval dt?

Boundary conditions and starting point

What is the formula used for the Simple Harmonic Oscillator Spring Force?

FSpring=kx

What three forces sum to zero in the Simple Harmonic Oscillator?

Spring Force, Drag Force, Inertia

Write the equation for the forces that sum to zero in the Simple Harmonic Oscillator.

𝑘𝑥 − 𝐷𝑉 + 𝑚𝑎 = 0

What is the resulting second order equation for the Simple Harmonic Oscillator?

𝑘𝑥 − 𝐷 𝑑𝑥 𝑑𝑡 + 𝑚 𝑑2𝑥 𝑑𝑡2 = 0

What is the definition of an Attractor?

Plotting the position and velocity of the mass as time advances generates an image known as the attractor which shows all possible states of the system.

What is the key idea behind the Euler-Cromer method?

Keeps the energy in the system exactly constant.

How does the Euler method compare to the Euler-Cromer method in terms of energy conservation?

The cumulative error in the Euler method adds energy to the system, while the Euler-Cromer method keeps the energy in the system exactly constant.

What are the components of the Pendulum Moment?

Fd, l

Write the equation for linear driving force

(A sin(Bt)) d^2theta/dt^2 = -(g/l)sin(theta)-(D/l)(d(theta)/dt) + (A sin(Bt))

What is the equation for a non-linear driving force?

(A sin(Bt)) d^2theta/dt^2 = -(g/l)sin(theta)-(D/l)(d(theta)/dt) + (A sin(Bt))

List pendulum parameters to create stable, predictable motion

Pendulum +10.0fabs(sin(t1.70))

List pendulum parameters to create chaotic, unpredictable motion

Pendulum +10.0fabs(sin(t1.70))

What are coupled oscillators?

l1 l2 K 1 2 K=Coupling (spring constant R/Nm)

How is the Earth and Sun modeled?

As a two dimensional problem

What equation represents the X component of the force between the earth and the sun?

cos( ) 2 2 2 r G M M dt d x M S E E = −

What equation represents the Y component of the force between the earth and the sun?

sin ( ) 2 2 2 r GM M dt d y M S E E = −