## Extra Credit 9

Explore two simulations.

### 1.  The pendulum lab

The interface

• If you want to do an experiment, "Pause" the simulation, set up your experiment, then start it.
• If you want to compare what happens when you change a variable, such as the length, check "Show 2nd Pendulum", "Pause" the simulation, set up your experiment, then start it.
• The "Photogate Timer" operates as a triggered mechanism, which starts when the pendulum crosses the vertical dotted line.  The period will be displayed after one cycle.
• Most objects can be dragged, the timer, the stop watch, the ruler, the tape measure, the energy graph and the entire green control panel.
• The initial angle is marked by a tick mark with the color of the pendulum mass.
• As you move the pendulum, the angles are constrained to be exactly integer number of degrees.

Make the following measurements and submit your answers to Questions 1 - 3 on Canvas.

Let the length of the pendulum be 1.5 m and the mass be 0.5 kg.  Choose no friction.  The formula for the period of a pendulum derived in the notes is T = (2π) (L/g)1/2.

• Calculate the expected period of the pendulum.
• Start the pendulum on Earth with an amplitude of 5 degrees.  Measure the period.  Is the measured period equal to the calculated period?
• Start the pendulum on Earth with an amplitude of 30 degrees.  Measure the period.  Is the measured period equal to the calculated period?
• Repeat everything with the same pendulum length but a 2 kg mass.
• Introduce damping by turning on friction and monitor the period.  Measure it several times before the motion damps out.
• Compare the periods of the same undamped pendulum on Earth, Moon, Jupiter, Planet X, and in a zero g environment.
• Devise a way to determine gX, the gravitational acceleration near the surface of planet X.

#### Question 1:

For the 2 kg mass, what is the ratio of the measured period with a 60o amplitude to the the ratio of the measured period with a 5o?

#### Question 2:

Which of the statements below is wrong?
a.  For the same amplitude and the same pendulum length, the period is independent of the mass.
b.  The period of a simple pendulum noticeably depends on the amplitude for large amplitudes.
c.  The net force acting on the pendulum bob is F= -mgsinθ.  The period increases with amplitude because this force increases faster than  harmonic restoring force F= -mgθ.
d.  A damped pendulum does not have a constant period.
e.  In a zero g environment a pendulum does not execute oscillatory motion.

#### Question 3:

What is gX, the gravitational acceleration near the surface of planet X, in units of m/s2?

### 2.  Resonances

The interface

• The "resonator" in the simulation is particular mass-spring system.
• If there is more than one resonator, a pull-down menu of preset combinations appears in the control panel on the right.
• The "Resonator" panel displays information about a particular "selected" resonator.  To select a resonator, either click on the blue mass or enter the resonator number in the text box next to the word "Resonator".
• You can change the mass and spring constant of the selected resonator with the sliders in the "Resonator" panel.  If you change any of the masses or springs, the pull-down menu box changes to "Select Preset". The "mixed m and k" choice is the default combination.
• There are two different frequencies displayed on the screen at any time.  The frequency in the "Resonator" panel is the natural frequency of the selected resonator.  The frequency on rotary control on the gray driver box is the driving frequency.
• The frequency dial can be turned through more than 360o.  The reason for this is to allow the user more precision in setting the exact frequency.
You can "Pause" the simulation and then use Step to incrementally see the motion.
• The "Ruler" comes with horizontal reference lines.
• To see the effects of varying the frequency and amplitude of the driver, it may be more instructive to use several resonators.

Make the following measurements and submit your answers to Questions 4 and 5 on Canvas.

A driven, damped harmonic oscillator with a harmonic driving force F = F0cosωextt will execute simple harmonic motion after a steady state has been reached.  Then the position varies as a function of time as

x(t) = Acos(ωextt + φ).

Here ω = ωext is is the angular frequency of the driving force.  The amplitude of the motion is given by

A = (F0/m)/[(ω02 - ω2)2 + (bω/m)2]½,

where ω02 = k/m.  ω0 is the natural frequency of the undamped oscillator.
When the driving force or the damping constant change, it takes some time before the new steady state is reached.  During that time interval the behavior is transient and the oscillator does not execute simple harmonic motion.

• Reset the simulation.  Change the spring constant to 400 N/m and the damping constant to 2 Ns/m, so that the natural frequency of the resonator is 2 Hz.  Set the driving frequency to 2 Hz.  Adjust the driver amplitude so that the amplitude of the motion is 10 cm.  (Use the ruler and the reference lines.)  When steady state has been reached, adjust the driving frequency upward until the amplitude of the motion is 5 cm.  (Everything else remains unchanged.)  Record the new driving frequency.  (Be patient, transient behavior can last for a long time.)
• When steady state has been reached, adjust the driving frequency downward until the amplitude of the motion is 5 cm.  Record the new driving frequency.
• Find ∆f/f, the fractional range of frequencies over which the resonator oscillates with at least half its maximum amplitude?

#### Question 4:

What is ∆f/f, the fractional range of frequencies over which the resonator oscillates with at least half its maximum amplitude in the above exercise (damping constant to 2 Ns/m)?

• Repeat the above exercise with a damping constant of 4 Ns/m.
• Does what you see in this simulation agree with what you understand from reading the notes?

#### Question 5:

What happens to ∆f/f, the fractional range frequencies over which the resonator oscillates with at least half its maximum amplitude when the damping constant changes from 2 Ns/m to 1 Ns/m?

Now go to Canvas, Assignments, Extra Credit 9.  Answer questions 1 - 5.  You can submit twice and the highest score counts.