In this introductory course on Ordinary Differential Equations, we first provide basic terminologies on the theory of differential equations and then proceed to methods of solving various types of ordinary differential equations. We handle first order differential equations and then second order linear differential equations. We also discuss some related concrete mathematical modeling problems, which can be handled by the methods introduced in this course. The lecture is self contained. However, if necessary, you may consult any introductory level text on ordinary differential equations. For example, "Elementary Differential Equations and Boundary Value Problems by W. E. Boyce and R. C. DiPrima from John Wiley & Sons" is a good source for further study on the subject. The course is mainly delivered through video lectures. At the end of each module, there will be a quiz consisting of several problems related to the lecture of the week.
9-5 Pendulum
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Из курса от партнера Korea Advanced Institute of Science and Technology
Introduction to Ordinary Differential Equations
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Korea Advanced Institute of Science and Technology
64 оценки
Из урока
APPLICATIONS OF SECOND ORDER EQUATIONS
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Kwon, Kil HyunProfessor
Department of Mathematical Sciences
Finally, let us consider the problem of the pendulum, okay?
In which you'll be able to follow in the situation.
A mass M, suspending from the end of the rod of length L,
is swinging in a vertical plane.
And let theta of T be the angle,
measured in radian from the vertical with the right hand side to be positive.
Look at this picture.
Here you have a certain mass,
suspended from the rigid ceiling and make
a displacement in the right direction by the angle theta.
Then the length of the arc with
the vertical angle theta is equal to capital lambda is equal to L of theta.
There'll be now from the geometry,
this is the length of this displacement capital lambda,
which is equal to L times a theta,
because the theta is measure in radian.
On the other hand, the angular acceleration,
the A which is equal to the second derivative of this displacement,
the capital lambda, so A is equal to the capital lambda double prime.
So that will be equal to L times the theta double prime, that's the acceleration.
Then, divide the Newton's law of the motion to the force,
total force acting on the system is given by the N times A,
because A is equal to L times theta double prime,
the M times the A is equal to M times L times theta double prime,
which must be the same as the negative MG sine of theta,
that this right hand side is actually
the tangential component of the gravitational force W is equal to MG acting on the mass.
MG is acting to the mass in the direction opposite to the motion.
Let's look at the picture again.
Here you have a gravitational force,
W is a weight is equal to MG.
If this displacement angle is a theta then this angle is equal to theta.
So that we have another force,
tangential component of this gravitational force is MG times sine of theta,
that is this much.
But be careful that the direction of this,
the tangential component of the gravitational forces opposite to the motion.
So that F is equal to MA must to be equal to negative MG sine of theta.
That's the equation we get here.
Force to be compute,
the force F is equal to MA,
using this lambda is equal to L theta,
so that if we have M times L times and theta double prime,
then must be equal to negative of MG times A sine theta,
there is a tangential component of the gravitational force.
So, simplify this equation.
M is a common in both sides.
So divide us through M,
you are going to get theta double prime plus G of L and sine theta is equal to zero.
Second order, highly nonlinear differential equation.
When the absolute value of the angle is quite small,
then we know that sine theta can be approximated by a theta.
Because, from the calculus,
you know that limit X tends to zero of a sine X over X is equal to one.
And that means, sine X over X behaves like one for quite a small.
That's what I mean.
So if the absolute value of theta is quite small
than sine of theta is approximately theta.
So that instead of considering this nonlinear differential equation,
that's approximated by theta double prime plus G over L theta,
because a theta is a good approximation of sine of theta.
Now, this is the second order,
Constant coefficient homogeneous differential equation.
With the symbol Omega,
which is defined by the skillet of G over L,
you can be lighted as,
I said double prime plus Omega square times the theta is equal to zero.
That this equation, this a question,
we call it as a linearized equation of the original nonlinear differential equation.
Even though this is not the true governing equation,
but when the absolute value of theta is a quite small,
it will give us a good approximation of the pendulum motion.
And as you can see from this equation,
this is exactly the same as that differential equation.
Theta double prime plus omega skillet theta is equal to zero.
This differential equation is the same equation as
the equation for the spring master system for Pre- undamped Supreme Motion.
There is no external force and there is no damped pin.
And we know that the general solution of this differential equation is
theta is equals to C1 cosine Omega T plus
C2 sine Omega T. Which you can combine into
capital A times cosine omega T minus P.
With the capital P is skillet of C one squared plus a C2 squared.
That's the amplitude of this pendulum motion
and the P is to face angle and the C1 is given
by the Capital A times the cosine P and
the CT is equal to capital A times the sine of P, okay?
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