Harmonic Oscillator Solution: Check:. Use Mathematica (plot_cosine.nb file) Notice period:
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Transcript of Harmonic Oscillator Solution: Check:. Use Mathematica (plot_cosine.nb file) Notice period:
![Page 1: Harmonic Oscillator Solution: Check:. Use Mathematica (plot_cosine.nb file) Notice period:](https://reader035.fdocuments.us/reader035/viewer/2022081501/56649ca35503460f949628e5/html5/thumbnails/1.jpg)
Harmonic Oscillator
Solution:
xkxkF
m
kx
dt
xd
xkdt
xdm
222
2
2
2
with
)cos( tAx
Check: x
tAdt
xdtA
dt
dx)cos()sin( 2
2
2
![Page 2: Harmonic Oscillator Solution: Check:. Use Mathematica (plot_cosine.nb file) Notice period:](https://reader035.fdocuments.us/reader035/viewer/2022081501/56649ca35503460f949628e5/html5/thumbnails/2.jpg)
Use Mathematica (plot_cosine.nb file)
2
TNotice period:
![Page 3: Harmonic Oscillator Solution: Check:. Use Mathematica (plot_cosine.nb file) Notice period:](https://reader035.fdocuments.us/reader035/viewer/2022081501/56649ca35503460f949628e5/html5/thumbnails/3.jpg)
Take x=0 at equilibrium and equations work out identicallyas for the horizontal spring:
Solution:
xkF
)cos( tAx
xdt
xd 22
2
![Page 4: Harmonic Oscillator Solution: Check:. Use Mathematica (plot_cosine.nb file) Notice period:](https://reader035.fdocuments.us/reader035/viewer/2022081501/56649ca35503460f949628e5/html5/thumbnails/4.jpg)
Energy:
2
2
1xkVxkF
constant2
1
2
1 22
vmxk
KVE
![Page 5: Harmonic Oscillator Solution: Check:. Use Mathematica (plot_cosine.nb file) Notice period:](https://reader035.fdocuments.us/reader035/viewer/2022081501/56649ca35503460f949628e5/html5/thumbnails/5.jpg)
Harmonic oscillators everywhere!
cos
sin
y
x
mgTF
mgF
2
2
2
2
dt
dL
dt
xd
Lx
L
g
dt
dmg
dt
dmL
2
2
2
2
1for sin
Pendulum
![Page 6: Harmonic Oscillator Solution: Check:. Use Mathematica (plot_cosine.nb file) Notice period:](https://reader035.fdocuments.us/reader035/viewer/2022081501/56649ca35503460f949628e5/html5/thumbnails/6.jpg)
Harmonic oscillators everywhere!
Level in connecting vessels
A floating bob
Attraction between atoms in a molecule
The charge in an LC circuit
![Page 7: Harmonic Oscillator Solution: Check:. Use Mathematica (plot_cosine.nb file) Notice period:](https://reader035.fdocuments.us/reader035/viewer/2022081501/56649ca35503460f949628e5/html5/thumbnails/7.jpg)
If you double the amplitude of a harmonic oscillator, the oscillator’s period
A. Decreases by 2
B. Decreases by √2
C. Does not change
D. Increases by √2
E. Increases by 2
![Page 8: Harmonic Oscillator Solution: Check:. Use Mathematica (plot_cosine.nb file) Notice period:](https://reader035.fdocuments.us/reader035/viewer/2022081501/56649ca35503460f949628e5/html5/thumbnails/8.jpg)
If you double the amplitude of a harmonic oscillator, the oscillator’s period
A. Decreases by 2
B. Decreases by √2
C. Does not change
D. Increases by √2
E. Increases by 2
![Page 9: Harmonic Oscillator Solution: Check:. Use Mathematica (plot_cosine.nb file) Notice period:](https://reader035.fdocuments.us/reader035/viewer/2022081501/56649ca35503460f949628e5/html5/thumbnails/9.jpg)
If you double the amplitude of a harmonic oscillator, the maximum speed
A. Decreases by 2
B. Decreases by √2
C. Does not change
D. Increases by √2
E. Increases by 2
![Page 10: Harmonic Oscillator Solution: Check:. Use Mathematica (plot_cosine.nb file) Notice period:](https://reader035.fdocuments.us/reader035/viewer/2022081501/56649ca35503460f949628e5/html5/thumbnails/10.jpg)
If you double the amplitude of a harmonic oscillator, the maximum speed
A. Decreases by 2
B. Decreases by √2
C. Does not change
D. Increases by √2
E. Increases by 2
![Page 11: Harmonic Oscillator Solution: Check:. Use Mathematica (plot_cosine.nb file) Notice period:](https://reader035.fdocuments.us/reader035/viewer/2022081501/56649ca35503460f949628e5/html5/thumbnails/11.jpg)
If you double the spring constant of a harmonic oscillator, the oscillation frequency:
A. Decreases by 2
B. Decreases by √2
C. Does not change
D. Increases by √2
E. Increases by 2
![Page 12: Harmonic Oscillator Solution: Check:. Use Mathematica (plot_cosine.nb file) Notice period:](https://reader035.fdocuments.us/reader035/viewer/2022081501/56649ca35503460f949628e5/html5/thumbnails/12.jpg)
If you double the spring constant of a harmonic oscillator, the oscillation frequency:
A. Decreases by 2
B. Decreases by √2
C. Does not change
D. Increases by √2
E. Increases by 2
![Page 13: Harmonic Oscillator Solution: Check:. Use Mathematica (plot_cosine.nb file) Notice period:](https://reader035.fdocuments.us/reader035/viewer/2022081501/56649ca35503460f949628e5/html5/thumbnails/13.jpg)
What will be the natural oscillation period of a 30-kg child on a swing whose seat is 3.2 m below the bar where the chains from the seat are attached?
A. Approx 0.5 sec
B. Approx 1 sec
C. Approx 4 sec
D. Approx 10 sec
E. Approx 40 sec
![Page 14: Harmonic Oscillator Solution: Check:. Use Mathematica (plot_cosine.nb file) Notice period:](https://reader035.fdocuments.us/reader035/viewer/2022081501/56649ca35503460f949628e5/html5/thumbnails/14.jpg)
What will be the natural oscillation period of a 30-kg child on a swing whose seat is 3.2 m below the bar where the chains from the seat are attached?
A. Approx 0.5 sec
B. Approx 1 sec
C. Approx 4 sec
D. Approx 10 sec
E. Approx 40 sec
![Page 15: Harmonic Oscillator Solution: Check:. Use Mathematica (plot_cosine.nb file) Notice period:](https://reader035.fdocuments.us/reader035/viewer/2022081501/56649ca35503460f949628e5/html5/thumbnails/15.jpg)
How do you measure the mass of an astronaut in orbit? NASA engineers designed a Body Mass Measuring Device. It is essentially a chair of mass m=20 kg mounted on a spring. (The other end is connected to the massive space station itself.) The period of oscillation of the empty chair is measured to be 1 s. When an astronaut is sitting in the chair, the period is 2 s. What is the mass of the astronaut?
A. Approx 20 kgB. Approx 40 kgC. Approx 60 kgD. Approx 80 kgE. None of the other options
![Page 16: Harmonic Oscillator Solution: Check:. Use Mathematica (plot_cosine.nb file) Notice period:](https://reader035.fdocuments.us/reader035/viewer/2022081501/56649ca35503460f949628e5/html5/thumbnails/16.jpg)
How do you measure the mass of an astronaut in orbit? NASA engineers designed a Body Mass Measuring Device. It is essentially a chair of mass m=20 kg mounted on a spring. (The other end is connected to the massive space station itself.) The period of oscillation of the empty chair is measured to be 1 s. When an astronaut is sitting in the chair, the period is 2 s. What is the mass of the astronaut?
A. Approx 20 kgB. Approx 40 kgC. Approx 60 kgD. Approx 80 kgE. None of the other options
![Page 17: Harmonic Oscillator Solution: Check:. Use Mathematica (plot_cosine.nb file) Notice period:](https://reader035.fdocuments.us/reader035/viewer/2022081501/56649ca35503460f949628e5/html5/thumbnails/17.jpg)
Known mobster and gambler Larry the Loser is found dead, hanging at about the 7-story level from a bungee cord tied to the top of the 15-story (45 m) Prudential Building in downtown Chicago. Larry has a mass of 90 kg and when the bungee cord is cut down, it is found to have a relaxed length of 5 stories. Detective Lestrade from the Chicago Police Department thinks that Larry must have been bungee-jumping from the top of the building and just unluckily hit the ground, but you know that Larry was murdered. What is your evidence?