Reading Quiz

Question 1:

Describe the phenomenon of resonance, give a real-life example, and explain why it is important.
  • Resonance the the phenomenon that describes the method by which repeated small applications of force close to the natural frequency can be used to generate large amplitudes. Resonance can be seen in airplane wings and on the swing set, making it very important to consider to explain the movement of objects under certain conditions.
  • Resonance is when the amplitude of an oscillation is made very large by repeated applications of a small force. This increased amplitude occurs close to the natural frequency (although in actuality the peak frequency is slightly lower than the natural frequency, which I did not understand the reasoning for). An example of this is pushing a swing or when the Tacoma Narrows bridge collapsed. It is important to consider resonance both in cases where you want to prevent large amplitude oscillations and you want to make them occur. When you want a large amplitude resonance allows you to create than amplitude with relatively little force, thereby expending less energy. On the other hand it is important to avoid building things, such as bridges, with certain resonant frequencies in order to prevent them from collapsing.
  • We say that an object in simple harmonic motion exhibits resonance when both the maximum amplitude increases as the driving frequency of the system approaches the natural frequency of the system, where the max amplitude is found at wm, which is approximated to wo in most cases, and when the phase lag between the position of the object driving the system and the position of the system itself jumps up to 90˚ as the driving frequency of the system once again approaches the natural frequency of the system, with the phase lag equaling 90˚ when the driving frequency equals the natural frequency. An example of this system is a "Texas tower", as shown in the book. Last semester Prof. Utter showed us a video of a bridge collapsing since the driving frequency of the wind lashing against it was close enough to the nat. freq. that the amplitude of the bridge's oscillations caused it to fall apart. Needless to say, it is important to understand the principle of resonance whenever you're dealing with an oscillating system,
  • when an object vibrates at its natural frequency. for example glass will shatter if vibrated at its natural frequency. this is important because if you are an opera singer and you're practicing for a big show, you need to know how not to shatter your nice drinking glasses.
  • Resonance happens when an oscillator (either damped or undamped) is also driven at a frequency that is roughly equal to the natural frequency of the oscillation. When this happens, the amplitude of the oscillation gets bigger, even when the force is small. A real life example is pushing a person on a swing (or pumping your legs). This is important because everything can resonate if it is forced at the right frequency, like buildings and bridges, which can compromise their structural integrity.
  • Resonance is when a driving force is applied to an oscillator near its natural frequency. This causes the amplitude to become huge. It can happen when a singer sings to break a wine glass. It is important because in physical machines such as cars there are things that happen periodically and if they cause resonance they can put huge stress on things and break them.
  • Resonance means that a vibrating system or external force drives another system to oscillate at greater amplitude at specific preferential frequency. For example, the phenomenon of resonance is widely used in musical instruments, because the instrument builders need to ensure that the resonance frequency is within the range of humans' ability of hearing.

    • Question 2:

    Describe what is meant by a coupled oscillator, and give an example (not already discussed in class).

    Answer:

    A coupled oscillator is made of at least two objects capable of exhibiting simple harmonic motion. However, their motion is coupled to each other through some physical means. Such an example is a swing-set with two swings; as one person stands swinging, motion begin to be 'transfered' to the other, initially stationary.
    1. A coupled oscillator is the result of tying the the motion of two oscillating objects together, this can include any extended object as every atom can be treated as an oscillator.
    2. Coupled oscillators do not oscillate independently; instead the movement of one oscillating object depends on the movement of the other. The most basic coupled oscillator is an atom or molecule, who's motions effect the other atoms and molecules around it. A more visually accessible example of a coupled oscillator is two rigid pendulums attached to a horizontal torsional rod.
    3. Coupled oscillators are oscillators that are 'attached' or so configured so that their oscillations are, in part, dependent on the oscillations of the other oscillators that it is 'coupled' to. An example given in the textbook is a pair of pendulums attached by a spring.
    4. its a system of masses connected by springs. we learned about these in 221, where a system has n-1 normal modes. you can think of any object as a coupled oscillator, because the atoms that make up matter can be thought of as masses connected by springs.
    5. Two or more oscillators are coupled if their motion is somehow related- so one's oscillation determines the motion of another and vice versa.
    6. coupled oscillators are two things in shm motion that interact with each other to create a new system of oscillators when combined. An example is two people on a merry go round with swings that are holding hands and the merry go round is speeding up and slowing down periodically.
    7. A coupled oscillator means two or more oscillators are connected in a way that energy could be exchanged between them. An example that is beyond our class discussion could be the system that the moon and the earth are orbiting around each other.

    Question 3:

    Name the two characteristics that, together, define a "normal" mode of oscillation.

    Answer:

    When a system of N particles is oscillating in a "normal" mode, (1) all N individual oscillators are moving with SHM at the same (normal) frequency, and (2) the amplitude of oscillation for each particle remains constant. NB: Condition (2) does NOT mean that all N particles have the same amplitude as each other.
    1. All masses vibrate at the same frequency and each has a constant amplitude.
    2. A normal mode is defined as when both coupled oscillators (masses in the example) oscillate at the same frequency and each has a constant amplitude.
    3. the equations of motion for a normal mode are: Xa = C cos w0*t Xb = C cos w0*t Where w0 is the nat. frequency of motion of each of the oscillators, and C is the amplitude of oscillation. Note that both A and B are completely in phase with each other.
    4. where all parts of the system of masses move together at the same frequency. the fixed frequency of the natural resonance.
    5. Normal modes happen when all oscillators have the same frequency, and have a constant amplitude (though the amplitude doesn't have to be the same for each oscillator, as in the case of a string where some places don't move, but others have large amplitudes)
    6. Every point oscillates at the same frequency and the ratios of magnitudes of displacements at various points are connected
    7. A "normal" mode of oscillation happens when all parts of a system moves with sinusoidally with the same frequency and the same phase shift.

    Question 4:

    Describe a way to extract the natural (normal) modes of oscillation for an extended object.

    Answer:

    Systematically driving the extended object over a range of frequencies will reveal special cases of resonance. This will occur at every natural frequency.
    1. Normal modes can be found by using symmetry, when are the masses moving in the same direction, and when do the directly oppose.
    2. In order to extract the natural modes of oscillation for an extended object, one can think of the object as a large number of simple oscillators coupled together.
    3. assuming that the solutions for the normal modes look like the equations above, we can see of there are and values for w, C', and C that fit the diff. eq. d^2Xa/dt^2 + (w0^2 + wc^2)Xa - wc^2Xb = 0 and d^2Xb/dt^2 + wo^2 + wc^2)Xb - wc^2Xa = 0. By substituting our assumed answer into the equation, we get (-w^2 + w0^2 + wc^2)C - wc^2*C' = 0 and -wc^2*C + (-w^2 + w0^2 + wc^2)C' = 0. Assuming that C and C' are non-zero and not independent, we can say that wc^2/(-w^2 + w0^2 + wc^2) = (-w^2 + w0^2 +wc^2)/wc^2 if we set both equations equal to C/C'. From here we can gather that w^2 = w0^2 + wc^2 ± wc^2, giving us two solutions for w, w1= w0^2 and w2 = w0^2 + 2wc^2. For w = w1, C/C' = -1 and for w = w2, C/C' = +1, meaning we have successfully extracted the natural normal modes of oscillation.
    4. you can get the natural modes by taking n-1.
    5. For an extended object, we must treat each piece as its own oscillator-- but we know that the frequency is the same. Then we can try to solve the equation of motion for each piece separately.
    6. In normal modes the amplitudes of displacements will be the same. So take various starting positions for each node at positive and negative displacements and model the situation for each one
    7. Release the extended object and wait until it can freely oscillate at several fixed frequencies, and then use oscilloscope or other experimental methods to get these values which are so-called normal modes.

    Question 5:

    What concerns or questions do you have concerning the material we covered in class?

    Answer:

    Your responses below.
    1. Nothing of note, but I would like to spend some time on the superposition of normal modes.
    2. In the resonance section, I was a little bit confused by figure 4-3 and the concept that a pendulum with a frequency lower than the natural frequency would correspond to a greater length than the true length. In addition, I would like to go over figure 4-8 in more detail because I am not entirely sure I completely understood it. Finally in this section, I was confused by the constants of integration idea. I know you also mentioned this in class and I didn't really understand it then either. I haven't really dealt with that concept in math, or at least not explicitly. In the coupled oscillators section I was a little confused by the idea that the two pendula connected by a spring were oscillating at different frequencies, as discussed on page 122. To me it seemed like they would be oscillating at the same frequency. Also I would appreciate if we could go over how to figure out the signs of the forces acting on a coupled system (pg. 124-5). Finally I was not sure what the distinction between dependent and independent was in the general analytical approach. I thought that all coupled oscillators were dependent on each other, so I did not see why the discussion of dependence was necessary.
    3. I definitely want to spend more time talking about these natural modes of oscillation, as I can clearly see that we only scratched the surface of this stuff in 221, and I feel like the more I see it, the more this stuff will really sink in.
    4. physically what determines an objects natural frequency? like why do they vibrate the way they do? is it something about the specific material? or the mode by which sound travels? why do normal modes behave the way they do? why/how do they exist in the first place?
    5. I'd like to go over how to extract the normal modes of a non symmetrical object, I didn't understand that that much.
    6. no serious concerns
    7. I didn't quite get the whole part of explaining the graph of resonant frequency, so I expect we could spend some time discussing it in class.