| 1. Run the applet by pressing the
"Start" ("Empiaza") button. Familiarize yourself with the
operation of the applet. The applet is broken into three major parts. The first shows a
particle undergoing circular motion at a constant rate. The second shows the same particle
on a vertical spring. Finally, the third plots the time evolution of the system -- the
result is a sine wave. Consider the positions of the two particles at any
time during the motion. (You can press the "Pause" ("Pausa")
button to freeze the motion if you find this helpful.)
(a) What is the relationship between the displacement vectors of the two particles?
(b) What is the relationship between the velocity vectors of the two particles?
(c) Given your answers to parts (a) and (b), can you guess how the acceleration
vectors of the two particles are related?
(d) By examining your answers to the previous questions, summarize an
important way in which constant circular motion is related to simple harmonic motion.
2. In physics, we often use the term "angular frequency" (or
"angular speed") to describe the frequency of oscillation of a particle on a
spring, even though the particle is moving along a straight line. (We even use the same
notation, w, and the same units, radians per second.)
(a) By examining the applet as it is running, describe why we can apply the angular
frequency of the particle undergoing constant circular motion to the particle on the
spring.
3. Set the frequency f = 1. As the applet runs,
notice that the applet generates a plot of a sine wave on the far right.
(a) What do the horizontal and vertical axes of the plot represent in terms of
the motion of the two particles?
(b) Where in the motion of the particle on the spring is the speed a maximum?
(c) Where in the motion of the particle on the spring is the speed a minimum?
(d) What points on the sine wave refer to the minimum and maximum speeds? How do
the slopes compare at these points?
4. Now we will consider what happens when the values of the
following parameters are changed:
(a) The Amplitude
(i) What happens to the frequency of oscillation when the amplitude is changed?
(ii) Is this surprising? Why or why not?
(b) The Frequency
(i) What happens to the amplitude of oscillation when the frequency is changed?
(ii) Notice that the wavelength changes when the frequency changes.
If the frequency is doubled, does the wavelength get larger or smaller? By exactly how
much?
(c) The Initial Phase ("Fase Inicial")
(i) Describe how changing the phase affects the following (if at all):
- the amplitude,
- the wavelength,
- the frequency.
(ii) What is the physical significance of the phase in terms of the initial position of
the particle on the spring?
(iii) Which value of the phase turns the sine wave into a cosine wave? What does
this mean physically?
5. Now consider the particle on the spring.
(a) When it is in its equilibrium position (that is, at x = 0), what is the net force
acting on the particle? Therefore, what would be the acceleration at this
point?
(b) By considering the net force acting on the particle at various positions, state
where in the motion of the particle on the spring the acceleration is a
maximum.
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