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

Use Mathematica (plot_cosine.nb file)

2

TNotice period:

Take x=0 at equilibrium and equations work out identicallyas for the horizontal spring:

Solution:

xkF

)cos( tAx

xdt

xd 22

2

Energy:

2

2

1xkVxkF

constant2

1

2

1 22

vmxk

KVE

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

Harmonic oscillators everywhere!

Level in connecting vessels

A floating bob

Attraction between atoms in a molecule

The charge in an LC circuit

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

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

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

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

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

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

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

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

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

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

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?