Conceptual Questions (4 points each)

Questions 1 -- 6 refer to the following situation:

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

   (c) Its acceleration is greatest when the spring returns to its initial 6 cm position.

   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). Choice (c) describes acceleration at an endpoint, where displacement and acceleration are maximal.

2. Which one of the following expressions gives the displacement of the block from equilibrium as a function of time? Let x be in cm and t in seconds.

   (e)

   The displacement from equilibrium is given by , where A is the amplitude, or maximum displacement, , and is determined by the initial configuration. Here A is 4 cm, describing the initial compression distance of the spring, which returns toward the equilibrium point at 10 cm when it is released. works out to be . Since the spring starts in its maximally compressed state, x = -A at t=0, and must be such that . Thus .

3. What is the maximal speed of the block?

   (d)

   The maximal speed is , here .

4. At what time t is the spring first maximally stretched?

   (c) after one half period

   After release at maximal compression, the spring returns to the equilibrium position (reaching it after 1/4 period), overshoots then reaches the maximally stretched position (after 1/2 period), returns through equilibrium again (at 3/4 period), reaching the maximally compressed state again after one period. The full cycle takes one period.

5. Suppose the spring is damped so that it loses 1/3 of its amplitude per cycle. By what factor is its mechanical energy multiplied per cycle?

   (d) 4/9

   If the spring loses 1/3 of its amplitude per cycle, the amplitude after a cycle is 2/3 times the amplitude before the cycle: . Since the spring's mechanical energy is given by , proportional to the square of A, means that .

6. Suppose the same block is mounted vertically atop the same spring. By how much does the spring compress?

   (c) 1 cm

   The block atop the spring is supported by the spring force, balancing its weight. Thus mg = kx, where x is the spring compression

   

   
   
   

   Questions 7 and 8 refer to the following situation:

   

7. Which one of the following statements concerning this situation is false?

   (e) The volume of water displaced by block A is greater than that displaced by block B.

   For these floating blocks, the buoyant force balances the weight of the block, and is proportional to the volume of water displaced (since it's equal to the weight of water displaced). Thus a denser block of the same volume is heavier, and requires a greater buoyant force and displaces more water (by floating lower in the water). Since blocks A and B have equal weight and density, they experience the same buoyant force (b) thus displacing the same volume of water (not (e)). C, floating lower in the water than A, must be denser (a), heavier, with a greater buoyant force (d), due to the greater volume of water displaced (c).

8. Let , , and represent the pressure at the bottom of blocks A, B, and C respectively. Which one of the following statements is true?

   (c)

   This can be obtained two ways. First, this is a static situation, with . Thus the pressure increases directly proportional to depth. The block bottoms have depths , so . Second, we could find the pressure on the bottom of the block as the buoyant force pushing upward, divided by the area of the block's bottom face: . A and B have equal buoyant force, but , so . A and C have equal surface areas on the bottom, but since C is heavier, and , with this argument again giving .

   Questions 9 -- 11 are based on the following situation:

   Two golf carts have horns that emit sound with a frequency 380 Hz. While cart A remains stationary, cart B approaches it at a speed of 12.0 m/s, then parks a distance 0.45 m away. The speed of sound is 343 m/s.

9. While cart B is approaching, what answer best describes what the driver of cart A will hear?

   (d) Beats, with beat frequency 14 Hz

   Driver A hears both horns: his own at 380 Hz, and cart B's, which is Doppler shifted since the source is approaching. B's frequency is thus shifted to

   

   Since A hears two sound waves with slightly different frequencies, the best answer for what he hears is (d): beats, with a beat frequency of 14 Hz, the frequency difference between the two waves.

10. 
    

    

    (c) Golfers on line L will hear nothing (destructive interference), while golfers on line M will hear a loud horn (constructive interference).

    Note that the wavelength of these sound waves is . Signals from A and B have the same path length in reaching golfers on line L, equidistant between them. Signals from B travel an additional half wavelength in path length than signals from A, in reaching golfers on line M. Since the signals are initially out of phase, traveling the same path length to line L keeps them out of phase, so they destructively interfere. By traveling an additional half wavelength to line M, the out-of-phase signals are made in phase on arrival at line M, causing constructive interference. Thus golfers on line L hear nothing, and golfers on line M hear maximal volume.

11. The driver of cart B turns off his horn, and hears cart A's horn at a sound intensity level of 80 dB. He then drives away until cart A's horn sounds half as loud (that is, its sound intensity level drops by 10 dB). To what distance from A must he go?

    (b) 1.4 m

    Since the decibel level is given by , reducing to means

    

    using the fact that . Thus

    

    Thus, to reduce the decibel level by 10, the intensity must be reduced by a factor of 10. Since intensity is proportional to , increasing the distance r by a factor decreases the intensity by a factor of 10, as desired. Thus A must go to distance .

    Questions 12 through 14 refer to the following diagrams, which show standing waves on six strings of the same length and wave velocity.

    

12. Which standing waves could be formed on a string with both ends free (or an organ pipe with both ends open)?

    (c) c and e only

    A string with both ends free has standing waves which have antinodes at both ends; these are the standing waves shown in c and e.

13. Which of the standing waves shown creates a sound with the highest frequency?

    (e) either e or f

    Since the strings all have the same wave velocity , the highest frequency waves have lowest wavelength. Since all strings are the same length, the lowest wavelength waves are those in e and f, which fit 3 half wavelengths on the string, for .

14. If the standing wave shown in f has the frequency 768 Hz, what is the frequency of the standing wave shown in a?

    (a) 256 Hz

    Note that the wave in a, with wavelength 2L, has 3 times the wavelength of the wave in e. Thus the frequency of wave a is 1/3 the frequency of wave e, or 256 Hz.

Quantitative Problems (Point Value as Marked)

1. (16 points)
   
   
   1. At what rate does the wine level in the bottle go down; that is, what is the wine's speed at the top of the bottle?
   2. What is the gauge pressure at the top of the bottle?

   
   
   
   
   
   
   

   1. By continuity, . Thus

      

   2. As this is a dynamic situation (wine is flowing), we must use Bernouilli's equation to find the difference in pressures. This tells us that the quantity is the same at the top as at the opening. Note that the height h = 0 at the opening, and the tiny velocity worked out in (a) will give a truly negligible contribution to the quantity . Finally note that the opening, being open to the atmosphere, is at atmospheric pressure. Thus, equating on the top (left hand side below) and bottom (right hand side below) gives

      

      or

      

      Note this pressure difference is negative; the pressure at the top of the wine column is less than that at the bottom. So the bottle experiences a slight inward force (slight because this 1940 Pa difference is only about 1 percent of atmospheric pressure), which it is strong enough to oppose and withstand.

2. (28 points)
   

   A 500 Hz wave travels to the left 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 speed v of the wave?
   3. Write down a mathematical expression, of the form , for this sinusoidal wave. Be sure you've given numeric values (and indicated units) for A, a, b, and .
   4. Sketch the ``movie'' graph showing the displacement of the string particle at versus time. Be sure to label, with numeric values, both the motion's amplitude and its period.

   
   
   
   
   
   
   
   
   

   1. From the graph, the wave goes through 1 cycle of its down-and-up sinusoidal waveform between x=0 and x= 0.4 m, so m.
   2. 
   3. In the mathematical form , A is the amplitude, , (with positive sign because the wave moves to the left), and tells us where in the cosine cycle the displacement starts, with the argument of the cosine given in radians. Here we read off

      

      Noting that the cosine function looks like

      

      we see that our displacement, which looks like a negative sin function, begins 1/4 of the way through the cosine cycle, at . Putting all together gives

      

   4. The particle at undergoes simple harmonic motion with period and amplitude 0.3 m, from the graph shown. It starts at the shown displacement (about 0.2 m), and then experiences the displacements which the wave carries to it from the right, first oscillating down, then up. Thus its motion is as plotted in the box above, with A = 0.3 m and as shown.

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