WEBVTT
00:00.120 --> 00:03.900
This is Giancoli Answers
with Mr. Dychko.
00:03.900 --> 00:06.360
We know that position
of a harmonic oscillator is amplitude
00:06.360 --> 00:10.790
times cosine
times the angular velocity times time.
00:10.790 --> 00:16.200
And we compare that with the formula
we're given 0.650 times cosine of 8.4 t
00:16.200 --> 00:18.740
and we compare corresponding parts here.
00:18.740 --> 00:21.500
So, we have this factor in front
of the trig function is amplitude.
00:21.500 --> 00:24.780
So, the amplitude is 0.650 meters.
00:24.780 --> 00:28.400
And then for Barbie
and we can figure out the frequency
00:28.400 --> 00:31.780
by knowing that the angular velocity
is 2π times frequency
00:31.780 --> 00:36.120
2π radians times the number
of cycles per second
00:36.120 --> 00:37.980
2π radians per cycle
times cycles per second
00:37.980 --> 00:40.000
gives you radians per second.
00:40.000 --> 00:44.460
And anyway, we divide this
by 2π on both sides
00:44.460 --> 00:46.880
and you get frequency
as angular velocity divided by 2π.
00:46.880 --> 00:51.400
So, that's 8.40 radians per second
divided by 2π.
00:51.400 --> 00:56.980
And we know that 8.4
because compare the inside parts here.
00:56.980 --> 01:00.500
The factor multiplied
by t is the angular velocity.
01:00.500 --> 01:02.880
So, ω is 8.4.
01:02.880 --> 01:05.840
This works out to 1.34 hertz.
01:05.840 --> 01:11.100
And then finding the total energy
01:11.100 --> 01:14.360
we know is 1/2 times spring constant
times amplitude squared.
01:14.360 --> 01:18.120
And frequency is 1 over 2π
times square root k over m.
01:18.120 --> 01:20.980
And we're going to solve for k
using this formula.
01:20.980 --> 01:23.290
So, we'll multiply
both sides by 2π first
01:23.290 --> 01:26.560
and then we'll square both sides
and then multiply it by m.
01:26.560 --> 01:29.900
And we get spring constant is
4π squared f squared times m
01:29.900 --> 01:31.860
and we substitute that in for k.
01:31.860 --> 01:35.420
And we get total energy is 1/2
times 4π squared f squared
01:35.420 --> 01:37.400
times mass times amplitude squared.
01:37.400 --> 01:39.280
And the 1/2 times 4 makes 2
01:39.280 --> 01:41.840
2π squared f squared A squared times m.
01:41.840 --> 01:46.390
And then we have 2π squared
times 1.3369 hertz squared
01:46.390 --> 01:49.680
and then multiplied
by the amplitude, 0.56 meters, squared
01:49.680 --> 01:51.710
times by the mass, 1.15 kilograms.
01:51.710 --> 01:55.180
And that gives 17.1 joules
of total energy
01:55.180 --> 01:57.660
the potential energy at any point is
01:57.660 --> 02:00.760
1/2 k
times that position squared.
02:00.760 --> 02:05.120
And we just found out before that
k is 4π squared f squared m.
02:05.120 --> 02:06.720
So, we plug that in
and then this works out to
02:06.720 --> 02:09.700
2π squared f squared x squared times m
02:09.700 --> 02:13.320
and that's 2π squared
times the frequency squared
02:13.320 --> 02:18.320
times this position here
of 0.36 meters squared times the mass
02:18.320 --> 02:20.320
and that works out too.
02:21.120 --> 02:27.020
5.26 six joules.
02:27.020 --> 02:29.420
That's the potential energy at this point
02:29.420 --> 02:31.300
and the kinetic energy
with the total energy
02:31.300 --> 02:33.000
minus the potential energy
02:33.000 --> 02:40.900
and so that's 17.1417
minus this 5.2581 joules
02:40.900 --> 02:42.520
using lots of digits here
02:42.520 --> 02:45.720
because we want to avoid
intermediate rounding error.
02:45.720 --> 02:52.500
So, we'll take 17.1417
minus the answer in the previous line.
02:52.500 --> 02:57.160
And we get kinetic energy must be
about 11.9 joules.