00:01
Here we have a mass attached to a spring, attached to the ceiling.
00:05
We have a spring constant, mass, and the damping constant looks like this.
00:11
Our equation of motion is mx double dot plus bx dot plus kx, and that equals our driving frequency, which is eight sine of three t cosine three.
00:36
I can rewrite that right -hand side as four times the sine of six t, because we recognize sine three t cosine three t is just the sine of twice that angle, so it looks like that.
00:55
And if we write out this with the numbers, we get five x double dot plus two x dot plus 40x.
01:11
So that's our master equation.
01:26
Then we're only interested in the steady -state solution, so we don't care about the damped stuff that happens early on and goes away.
01:40
So we're gonna look for a solution of the form x is some amplitude times the sine of six t plus phi.
01:54
Phi is gonna be some phase angle, a is the amplitude.
02:00
And we can tell it's the six because of the six that's up here on the right -hand side.
02:08
So we get x dot is six a cosine, and x double dot is minus like that.
02:30
Then we substitute that into our equation.
02:34
Minus 180 times a times the sine, and that's equal to four sine of six t.
03:19
Okay, so then we break these sines and cosines up using the usual sine and cosine of a sum.
03:32
We'll make it so we get minus 140a, that's the sine term.
03:38
So we get sine of six t cosine phi plus cosine of six t sine of phi plus that's 140a, 12a times cosine, cosine minus sine...