Conceptual Questions

NOTE: For Questions 1 -- 3, circle ALL correct responses. More than one answer may be correct.

Questions 1 -- 3 refer to the following context:

Two loudspeakers are located 3 m apart on the stage of an auditorium. A listener at point P is seated 29 m from one speaker and 25 m from the other. The speed of sound is 343 .

Consider each of the following situations:

1. Both speakers are in phase, emitting a frequency of 113 Hz.
2. Both speakers are in phase, emitting a frequency of 43 Hz.
3. Both speakers are out of phase, emitting a frequency of 113 Hz.
4. Both speakers are out of phase, emitting a frequency of 43 Hz.
5. The speakers emit slightly different frequencies, of 100 and 102 Hz.
6. Both speakers emit a frequency of 113 Hz; the rightmost speaker glides on a track, approaching the listener with speed 4 .

Circle EACH of the following situations for which the listener hears

1. Constructive interference (a sinusoidal sound wave of maximal intensity)

   ONLY for (d) IV

   To obtain (wholly) constructive interference -- that is, a sinusoidal sound wave of maximal intensity -- the listener must receive both sound waves at the same frequency and in phase. This eliminates (V) and (VI), where the listener receives waves with different frequencies [in (VI), because the wave from the approaching speaker is Doppler shifted in frequency].

   For (I) through (IV), where the waves arrive with the same frequency, we must consider their relative phase. The difference in path length between the two received waves is 4 m, and this constitutes some multiple a of the wavelength. We relate this to the waves' frequency, the variable stated in our situation descriptions:

   

   plugging in the given path difference m and wave speed .

   Note that when the waves are emitted in phase, the path difference must be an integer number of wavelengths, a = n, to keep both waves in phase. When the waves are emitted out of phase, the path difference must be an odd half number of wavelengths, a = n + 1/2, to get both waves back into phase.

   We then note that the given frequencies correspond to a = 1.3 for (I) and (III) -- a path difference of 1.3 wavelengths -- and a = 0.5 for (II) and (IV) -- a path difference of one half wavelength. A path difference of 1.3 wavelengths will give wholly constructive interference in neither case (speakers in or out of phase), while a path difference of one half wavelength will give wholly constructive interference only when the speakers are out of phase. Thus (IV) is the only situation with wholly constructive interference.

2. Destructive interference (a sinusoidal sound wave of minimal intensity)

   ONLY for (b) II

   Wholly destructive interference occurs only when both sound waves are received with the same frequency, out of phase. This eliminates situations (V) and (VI), as above, since the waves arrive with different frequencies.

   When waves of equal frequency are emitted in phase, the path difference must be an odd half number of wavelengths, a = n + 1/2, to get the two waves out of phase. When the waves are emitted out of phase, the path difference must be an integer number of wavelengths, a = n, to keep the waves out of phase.

   As above, situations (I) and (III) involve a path difference of 1.3 wavelengths, while situations (II) and (IV) involve a path difference of one half wavelength. A path difference of 1.3 wavelengths will give wholly destructive interference in neither case (speakers in or out of phase), while a path difference of one half wavelength will give wholly destructive interference only when the speakers are in phase. Thus (II) is the only situation with wholly destructive interference.

3. Beats

   ONLY for (e) V and (f) VI

   Beats are obtained when the listener receives sound waves of differing, but close, frequencies. This occurs in situation (V). It also occurs in situation (VI), since the wave from the approaching speaker is Doppler shifted in frequency.

   
   

   NOTE: For the remaining multiple choice problems, choose the ONE best answer.

   Questions 4 through 8 refer to the following situation:

   

4. Which one of the following statements is true concerning the motion of the block ?

   (e) Its acceleration is greatest when the mass has reached its maximum displacement.

   The period depends only on k and m; velocity and acceleration vary sinusoidally, with velocity maximal at the center of the motion (zero displacement) and acceleration maximal at its endpoints (maximum displacement).

5. What is the maximum acceleration of the block?

   (e)

   The maximal acceleration is , where A is the amplitude and is the angular frequency of the oscillation. Here

   

   so

   

   using the amplitude given, .

6. What instantaneous speed does the hammer blow give to the block?

   (d)

   This could be found in two equivalent ways. Both require recognizing that the hammer blow gives the block an instantaneous velocity at its equilibrium position, where the block's potential energy is zero, so kinetic energy and speed are maximal. Thus we find

   

   or equivalently, setting our initial kinetic energy equal to potential energy at maximum displacement,

   

   as above.

7. Which of the following graphs BEST illustrates the kinetic energy versus time for the oscillating block-spring system?

   

   To answer this, note that we begin at equilibrium, with maximum speed and kinetic energy and that kinetic energy is always positive. This rules out all but graph II.

   For more detail, velocity here oscillates sinusoidally, starting at its maximally negative value (since it's going to the left). Kinetic energy looks like the square of that sinusoidal variation, hitting it's maximal value whenever v hits either plus or minus (that is, it peaks whenever v is at a crest or a trough). What is shown is the kinetic energy for one cycle of the spring's oscillation, as v goes from , through zero, to , and back again.

8. At what time t is the spring first maximally compressed?

   (b) after one quarter period

   The spring starts in the equilibrium position and begins compressing (the block displacing to the left). It's displacement thus looks like

   

   first reaching minus its amplitude (maximal compression) after one quarter period.

   Questions 9 and 10 are based on the following situation:

   
   

   Two boys build a log raft to sail down the river. The raft has volume 4.4 and density , while the river water has density .

9. What is the maximum weight (including themselves) that the boys can load onto the raft, without allowing the top of the raft to sink below the water's surface?

   (c) 8600 N

   

   Thus

   

   since the raft displaces its entire volume in water. Plugging in numbers,

   

10. Assume the raft, with height h = 0.3 m and area , is fully loaded as in the question above (that is, its top is level with the water surface). What is the gauge pressure at the bottom surface of the raft?

    (b)

    First, note that the top surface of the raft (and of the water) is at atmospheric pressure , and that gauge pressure measures .

    This question can be answered in two equivalent ways. First, note that the bottom of the raft is below the water surface. Thus the gauge pressure, at this water depth, is

    

    Note that it is , and not , determining the pressure, since the fluid has uniform pressure at any depth h, by Pascal's principle. This gives

    

    Alternatively, we know that the buoyant force upward is just the net upward force due to pressure on the bottom and top of the raft:

    

    Thus

    

    giving as above.

Quantitative Problems (Point Value as Marked)

1. (15 points)
   
   
   1. With what speed does the oil flow upward, at point P, 5 m below ground in the reservoir?
   2. What is the gauge pressure at point P?

   
   
   
   
   
   
   

   1. By continuity, taking oil to be incompressible, the volume rate of flow vA is constant. Thus

      

      giving .

   2. We use Bernoulli's equation, is constant, taking h to be zero at P and 5 m at the ground, 5 m above. (Note we must use Bernoulli's equation, as the fluid is flowing, not static.) Thus

      

      The gauge pressure at P is thus

      

      where we've discarded the kinetic energy term at P to simplify the calculation, since it is 1000 times smaller than terms we keep.

2. (30 points)
   

   A wave travels to the right with velocity 6 m/s along a string. Its displacement has the following ``snapshot'' at time t=0.

   1. What is the wavelength of the wave?
   2. What is the period T of the wave?
   3. Write down a mathematical expression, of the form , for this sinusoidal wave. Be sure you've specified A, a, b, and numerically.
   4. Sketch the ``movie'' showing the displacement of the string particle at versus time. Be sure to label, numerically, both the motion's amplitude and its period, and to start your graph at t= 0.

   
   
   
   
   
   
   
   
   

   1. From the graph, the wave goes through 1 cycle of its up-and-down sinusoidal waveform between x=0 and x= 0.2 m, so m.
   2. , so .
   3. In the mathematical form , A is the amplitude, , (the sign is because the wave moves to the right, with positive ), and tells us where in the cosine cycle the displacement starts. Here we read off

      

      Noting that the cosine function looks like

      

      we see that our displacement, which looks like a sin function, begin 3/4 of the way through the cosine cycle, at . Putting all together gives

      

   4. The particle at undergoes simple harmonic motion with period and amplitude 0.15 m, from the graph shown. It starts at the shown displacement (about 0.1 m), and then experiences the displacements which the wave carries to it from the left, first oscillating down, then up. Thus its motion is as plotted above, with A = 0.15 m and as shown.

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