0:11

By which I mean, there is no external force.

In other words, f is identical to 0.

And there is no damping, so that the damping constant c is equal to 0, okay?

Then the given differential equation becomes,

simpler one, x double prime + omega square times x = 0.

And this general solution is very easy to

find because the characteristic equation

corresponding to this differential

equation is r2 + omega squared = 0, so

that we have two roots for r given by +or- i times omega.

So that its general solution becomes x(t)

= arbitrary constant c sub 1 times cosine omega t,

and + another arbitrary constant c sub 2 times sine omega t, all right?

But using the trigonometrical identity,

we can combine these two terms into one as

capital A times cosine (omega t- phi),

where A = square root of c1 squared + c2 squared.

And the c1 = capital A times the cosine phi,

c2 = A times the sine phi, okay?

Here, we call the capital A, given by this quantity, the amplitude.

And the angle phi, the phase angle, of the simple harmonic motion (4).

We call this the equation.

The equation 4 has a simple harmonic motion and where we call the A,

the amplitude and phi the phrase angle, okay?

2:36

Let's look at the picture for the simple harmonic motion, okay?

It's a kind of cosine curve,okay?

Where we have the maximum altitude, capital A, this is amplitude, right?

2:51

And where the maximum values obtained when t

is equal to phi over omega, okay?

And from this, the length of the consecutive two maximum points,

that is equal to period 2 pi over omega, right?

That's the graph of the simple harmonic motion, okay?

For example, let's assume that we have

an 1kg mass is attached to a spring with

stiffness 4 kg per second square.

And at time t = 0, the mass is stretched downward scaled

to 3 over 4 meters from its equilibrium point and

then released with upward velocity one-half meters per second, okay?

Assuming there is no damping and no external force, find its motion and

the amplitude, and the phase angle in -pi and pi, okay?

4:14

And the problem says, the motion is a free undamped motion

because there is no external forces free and because we assumed

that there is no damping, so it's undamped too, right?

4:30

And m = 1, and the stiffness, k = 4, right?

So that we have an initial value problem, very simple one, x double prime + 4x = 0.

Initial displacement is the square root of 3 over 4.

And the initial velocity, because it is moving outward,

it should be- one-half, right?

Solving this very simple homogeneous second order,

constant coefficient differential equation.

With these two initial conditions,

you can easily get x(t) = square root of 3

over 4 cosine 2t- 1 over 4 sine 2t, okay?

And using the trigonometric identity, we can combine it

into a one-half times cosine 2t + pi over 6, right?

5:33

One-half amplitude you get the following.

This is equal to c1 squared, in other words,

the square root of 3 over 4 squared and + (- 1 over 4)2, right?

That's the amplitude one-half, okay?

5:51

And the angle is equal to,

you get A is equal to one-half times the cosine of phi,

that is equal square root of 3 over 4, and

one-half of sine of phi, that is equal to -1 over 4.

From this, you can get the phase angle,

phi is equal to- pi over 6.

6:19

The mass passes through the, that's the harmonium motion we obtain, okay?

And finally, to get the moment at which the mass passes

through the equilibrium position, and

equilibrium position means that the displacement x(t) = 0.

So let's put x(t), say,

one-half times cosine 2t + pi over 6 = 0,

then cosine is equal to 0 when the argument

is pi over 2 + k pi, where k is any integer.

So that we should have 2t + pi over 6 must be equal

to pi over 2 plus k pi, where k is an arbitrary integer.

7:10

So first such time, first such positive time is when k is equal to 0,

from this expression, you will get pi over 6, right?

So that means after pi over 6 seconds, the mass passes through

the equilibrium position, in other words, at which x = 0, right?