Basic Physics III Lecture 1
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Basic Physics III1 / 27
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00:00
NasspulvergießenParticle physicsJuneMassCaliberFile (tool)Linear motorSpeise <Technik>Multiplexed Analogue ComponentsHarmonicFaxFirearmNegativer WiderstandFree fallPhase angleAudio frequencyBeta particleH-alphaFahrgeschwindigkeitThermodynamic equilibriumAccelerationAmplitudeInitiator <Steuerungstechnik>Gas turbineOrbital periodForceSpring (season)Mitsubishi A6M ZeroRoots-type superchargerMassCosmic distance ladderCombined cycleCartridge (firearms)Ground (electricity)HarmonicContrast (vision)Capital shipNoise figureAtmosphere of EarthSeries and parallel circuitsMinerMaskMorse codeMeasurementPhase (matter)MarsSizingToolLastYearShip classVoltaic pileFlugbahnComputer animation
06:10
HarmonicThermodynamic equilibriumTiefdruckgebietForceLadungstrennungCosmic distance ladderColor chargeSpring (season)Negative feedbackMultiplexed Analogue ComponentsLocal Interconnect NetworkBecherwerkLinear motorParticle physicsHuePhotographic plateAudio frequencyMaskFahrgeschwindigkeitRing (jewellery)Phase (matter)LuggerTelevision setFuel economy in automobilesNut (hardware)ThursdayColor chargeFahrgeschwindigkeitCartridge (firearms)Direct currentForceSpring (season)Audio frequencyCosmic distance ladderBombAmmeterPhase angleAir compressorPhase (matter)AccelerationRack railwayThermodynamic equilibriumBauxitbergbauAmplitudeInitiator <Steuerungstechnik>ToolAtmosphere of EarthGroup delay and phase delayHang glidingFrictionCaliperMaskMarsOrder and disorder (physics)Will-o'-the-wispKilogramCrystal twinningBand gapMitsubishi A6M ZeroMicrometerMassCoulomb's law
12:20
Audio frequencyIsolationsüberwachungRad (unit)FahrgeschwindigkeitCrystal habitRing (jewellery)Mot (Semitic god)HarmonicPhase (matter)Thermodynamic equilibriumBomberAC power plugs and socketsPair productionAlcohol proofMinerAmmeterPhase angleBook designRemanenceHarmonicTitan Saturn System MissionSpare partCartridge (firearms)Orbital periodLecture/Conference
15:19
Cocktail party effectComputer animation
Transcript: English(auto-generated)
00:04
We will start our discussions with the phenomenon of oscillations or simple harmonic motion. Let me contrast simple harmonic motion with that of free fall under gravity.
00:23
If we have a free fall under gravity, there is a potential energy that increases linearly with the height on the horizontal axis. The force is downward and constant, so the force is minus the mass times gravity.
00:45
In contrast for simple harmonic motion, which in this case we use a mass attached to a spring, the potential energy is a quadratic function and there is a proportionality between the force and the deviation from the equilibrium position.
01:09
The equilibrium position is where this potential energy is smallest. If this denotes x0, then we measure the deviation from x0.
01:23
The force is proportional to that deviation with the proportionality constant of k or the spring constant. If we look at the corresponding motion for the free fall motion, it is a parabola, so the height changes with time like that.
01:45
The only thing that can happen is that it can bounce off the ground, but it just falls down. While the simple harmonic motion is a periodic motion, it looks something like this. We characterize this periodic motion around the equilibrium position x0 with the amplitude, which
02:08
is the maximum deviation from x0, so it's the distance from here to here, and then with the period T, which is the distance from the smallest
02:22
position to the smallest position or from the biggest position to the biggest position. Finally, our last constant is the phase angle phi, and that is the distance from the first maximum to the origin to time equals zero.
02:41
Here we have an actual system that does this free fall, so just a mass that is falling under gravity and then hitting the ground. This is the mass attached to the spring, and it oscillates back and forth like this.
03:02
To reiterate, here the potential energy is linear, here it is quadratic, the mass just falls down, the mass oscillates back and forth. The force here is constant, just minus mg, while the force here is proportional to the deviation from the equilibrium position x0.
03:26
Then from F equals ma, we get that the acceleration of the motion is minus g, and here we get a differential equation, meaning that the second derivative of the deviation from the equilibrium is proportional to that deviation itself,
03:44
and the proportionality constant is k over m, where k is the spring constant, the same as for the proportionality constant for the force, and m is the mass.
04:00
So, the resulting motion here is this parabola, which has the initial condition of the initial height here and the initial velocity, and the curvature is minus g over 2 times the time squared, while here we have a combination of cosine and sine functions,
04:22
and there are also initial conditions, the initial deviation from the equilibrium ds and the initial velocity vs. And then we can get from the position or from the height here, we can get the velocities and the accelerations.
04:41
If we now look at those initial conditions, so let's say that at time equals 0, we have an initial deviation from the equilibrium of ds and an initial velocity of vs, and we define now the amplitude in terms of that initial condition, and that amplitude is defined as
05:07
the square root of the initial deviation from equilibrium squared plus m over k times the initial velocity squared. And we define the angular frequency omega, which is 2 pi times the frequency or 2
05:27
pi over the period, and omega squared is given as the ratio of k over m. And we define the phase angle phi by these conditions, that the amplitude times the cosine equals the initial deviation from equilibrium position,
05:46
and minus a times the sine of that phase angle is the initial velocity divided by omega. And we now use cosine alpha plus beta equals cosine alpha cosine beta minus sine alpha sine beta, so that's a trigonometric identity.
06:06
And we can then use those definitions and rewrite the position as a function of time as well as velocity and acceleration like so. So the position is the equilibrium position plus a times cosine omega t plus phi, and the velocity is minus a omega times sine omega t plus phi,
06:34
and the acceleration is minus a omega squared times cosine omega t plus phi.
06:42
And the minimum position is x naught minus a, and the maximum position is x naught plus a. And the minimum velocity is minus a omega, and the maximum velocity is plus a omega. The minimum acceleration is minus a k over m, and the maximum acceleration is a k over m.
07:07
We have used that omega squared equals k over m. So let's look at an example. Let's look at a DNA molecule, which is 2.17 micrometers long. And a DNA molecule, because of its double helix structure, can act like a spring.
07:26
And so let's say the ends of the molecule becomes singly ionized, meaning a negative elementary charge on one end and a positive elementary charge on the other. And so because of the attraction of the negative and the positive elementary charge, the double
07:46
helix compresses by 1%, and we are going to determine the spring constant of the molecule. So we know the equilibrium position of the DNA molecule. It's 2.17 times 10 to the minus 6 meters, so 1 micrometers, 10 to the minus 6 meters.
08:05
And the deformation or compression is x minus x0 is 1% of that equilibrium length. So in other words, x minus x naught is 2.17 times 10 to the minus 8 meters.
08:25
So not times 10 to the minus 6, but times 10 to the minus 8, and that is this factor of 1%. Therefore the distance between the two charges is no longer 2.17 times 10 to the minus 6, but it is slightly smaller, so it is 2.15 times 10 to the minus 6.
08:48
The electric force between the two charges can be calculated as 1 over 4 pi epsilon naught times the charges squared over the distances squared.
09:02
And 1 over 4 pi epsilon naught is 8.99 times 10 to the 9 Newton meters squared over Coulomb squared. Now that we have both the deviation from equilibrium as well as the force, we can calculate the spring constant K as the ratio of those two.
09:22
So it is the 4.99 times 10 to the minus 17 Newtons divided by this 2.17 times 10 to the minus 8 meters, and that is 2.3 times 10 to the minus 9 Newtons per meter. So let's say we have given a spring, and we know that if we hang a mass from it, a mass of 0.3 kilogram, it stretches by 0.15 meters.
09:51
And it is attached vertically originally, and the first problem is to determine the spring constant, and we do it in the same way as we did for the DNA molecule.
10:01
So we know the force that stretches the spring is m times g, and we know that the spring is stretched by 0.15 meters. So we have 0.3 kilograms times 9.8 meters per second squared divided by 0.15 meters, and that gives us 19.6 Newtons per meter.
10:35
So if we now take off that mass and mount it horizontally, so we attach it horizontally now and it glides without friction horizontally.
10:45
Now that we know the spring constant and the mass, we can determine the angular frequency using omega squared equals k over m. So that means that omega squared, plugging in the numbers, is 65.3 per second squared, and that means that omega itself is 8.08 per second.
11:08
Now the spring is first compressed by 0.1 meters from equilibrium, and that means x of 0 minus x naught is minus 0.1 meters.
11:21
And in addition, it is given a shelf to create a velocity at time equals 0, or vs of 0.4 meters per second in the positive x direction. Now we are going to determine the amplitude, frequency, and the phase of the simple harmonic motion.
11:41
So we use the relation for the amplitude first. We know that the amplitude squared is the sum of the initial deviation squared, the 0.1 meters, plus the initial velocity divided by the angular frequency. So that's the 0.4 meters per second divided by 8.08 per second, and then that's squared.
12:07
And from that we get the amplitude squared as this, or the amplitude itself as 0.112 meters. So for the phase we need to use the sine and the cosine. To get the phase angle we use the two equations x minus x naught equals a cosine omega t plus phi, and v equals minus a omega sine omega t plus phi.
12:51
And because tangent x equals sine x over cosine x, we can take the ratio of those two equations.
13:05
So we get v of 0 over x of 0 minus x naught equals minus a omega sine 0 plus phi over plus a cosine 0 plus phi.
13:33
And the a cancels, and we're just left with the omega and the tangent function of phi.
13:42
So we get minus omega tangent of phi. Now when we actually plug in the numbers, the tangent function has an ambiguity or periodicity. So the tangent function looks roughly like this.
14:10
So it repeats with the periodicity of 180 degrees or pi in radians.
14:23
And therefore the calculator will just give us the answer for tangent phi using this piece of the tangent function. But we don't know whether it is on this one, on this one, or on that one.
14:40
So we simply have to try out those original equations and see which one fits best. And it turns out in this case we need to add 180 degrees or pi. And so with that we find that the phase angle is 3.6 radians or 206.3 degrees.
15:01
And we can then write down the equation for the simple harmonic motion as x of t equals x naught plus 0.112 meters times cosine 8.08t per second plus 3.60. Thank you for your attention.