Extra Credit 9

Explore two simulations.

1.  The pendulum lab

Link to the simulation:  https://phet.colorado.edu/en/simulation/legacy/pendulum-lab
Click "download" or "Run Now!"

The interface

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.

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

Link to the simulation:  http://phet.colorado.edu/en/simulation/resonance

Click "download" or "Run Now!".

The interface

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.

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)? 

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.