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    Various Practice Problems

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    Physics questions

    The expression given below represents a transverse harmonic wave travelling along a
    string which is stretched along the horizontal x-axis. The expression yields the transverse
    wave displacement y in metres when x is entered in metres and the time t in seconds.
    For 1 mark each, answer the following questions (be sure to give physical units with your
    answers):
    [2] (a) What are the frequency and wavelength of this wave?
    [1] (b) What is the propagation speed c of this wave?
    [1] (c) In what direction is this wave travelling?
    [2] (d) What is the maximum transverse velocity achieved by any particle of the string?
    ?
    y(x,t) = .05 sin(60t + 2x +1)
    Phys 321 Module 10 Exam your ID
    Fall, 2002 page 3 of 10
    continued.....
    A2. On a winter day at a location near Calgary, a
    layer of cold Arctic air at -40°C occupies a
    valley bottom. Above the cold air is a layer of
    warmer Pacific air at +20°C. To a good
    approximation, the air pressure in both layers
    can be assumed to be the same and equal to
    8.00x104 Pa.
    [3] (a) Find the numerical value of the ratio of acoustic impedances of the two layers. Which
    has the higher impedance? (Recall gas density is inversely proportional to temperature at
    constant pressure.)
    [1] (b) A sound wave is beamed vertically upward towards the horizontal interface between
    the cold air and the warmer air. Is the reflection coefficient at the interface positive
    or negative? Explain very briefly.
    [2] (c) What fraction of the energy carried by the sound wave of part (b) is reflected from the
    interface?
    -40° C
    +20° C
    Phys 321 Module 10 Exam your ID
    Fall, 2002 page 4 of 10
    continued.....
    A3. An isotropic point source of sound waves produces an intensity level of 83.0 dB at a
    distance of 1.00!m from the source. The source is surrounded by a uniform medium
    consisting of air at 20°C with sound speed 344 m/s.
    [2] (a) What is the intensity level (in dB) 1000 m from the source, assuming absorption of
    0.01 dB/m in the surrounding air?
    [2] (b) What is the acoustic intensity (in W/m2) at a distance of 1000 m from the source?
    [2] (c) Calculate the total power being emitted by the source at any instant.
    (Iref for the decibel scale is 1x10-12 W/m2.)
    Phys 321 Module 10 Exam your ID
    Fall, 2002 page 5 of 10
    continued.....
    A4. The diagram shows a plot of a triangular transverse wave pulse as a function of x at t=0.
    Note that the vertical scale on the plot is different from the horizontal scale. The pulse is
    moving to the right on a string under tension 600 N at wave speed c=40 m/s.
    Using the axes provided, plot the transverse particle velocity vy and the transverse force
    Fy for this pulse as functions of x at t=0. Be careful to label your vertical axes with
    appropriate scales. Scales should include physical units.
    [3]
    [3]
    y(x,0)
    vy(x,0)
    P(x,0)
    x
    x
    x
    -2 m -1 m 0 +1 m
    -2 m -1 m 0 +1 m
    -2 m -1 m 0 +1 m
    5 cm
    Fy(x,0)
    40 m/s
    Phys 321 Module 10 Exam your ID
    Fall, 2002 page 6 of 10
    continued.....
    A5. A liquid has bulk modulus 9.1x108 Pa and density 0.81x103 kg/m3. A plane harmonic
    acoustic travelling wave of frequency 1000 Hz creates an acoustic intensity of 1.00x10-6 W/m2
    at a point P in the liquid.
    [1.5] (a) What is the speed of acoustic waves in this liquid?
    [1.5] (b) What is the acoustic impedance of this liquid?
    (c) At an instant of time when the velocity of the fluid element at P is zero, what are
    [1.5] (i) the pressure fluctuation at P?
    [1.5] (ii) the absolute value of the displacement of the fluid element at P?
    Phys 321 Module 10 Exam your ID
    Fall, 2002 page 7 of 10
    continued.....
    A6. A guitar string has mass per unit length 3.00 grams/metre and is stretched under a
    tension of 1500N with its ends fixed a distance 80.0 cm apart.
    [2] (a) What is the frequency of the fundamental mode of vibration (i.e., the 1st harmonic) of the
    string under these conditions?
    [2] (b) Suppose the x-axis below lies along the guitar string, with the fixed ends at x=0 and
    x=80 cm. Sketch the shape of the displacement y of the string when it is vibrating in its
    second overtone (i.e., 3rd harmonic) under the given tension. Assume the particle at
    x=20!cm has a negative displacement, as indicated.
    [2] (c) Referring to the situation you drew in part (b), suppose that the displacement of the
    particle at x=20 cm is -0.800 mm at the instant depicted in the figure. What then is the
    maximum displacement to be found anywhere along the guitar string at this same instant?
    x
    y
    0 20 cm 40 cm 60 cm 80 cm
    particle at x=20 cm
    fixed end fixed end
    Phys 321 Module 10 Exam your ID
    Fall, 2002 page 8 of 10
    continued.....
    y(x,0)
    x
    1 m/s
    -1 m 0 m !2 m
    y = 1 m
    Part B. [9 marks] Attempt ONE of the two problems below. Use the
    following two blank pages for your answer.
    B7. The transverse wave pulse shown below travels on a stretched string in the positive xdirection
    at wave propagation speed 1 m/s. The tension in the string is 1 N. The
    displacement y of the pulse at t=0 is given by the function to the right of the figure. This
    function returns the value of y in metres when x is entered in metres.
    [5] (a) Find the particle velocity at x="2 m as a function of time, i.e., find vy("2,t). Draw a
    reasonably accurate plot of vy("2,t) versus t.
    [4] (b) Find the total energy (in joules) being transported by this wave pulse.
    OR
    B8. Two loudspeakers, A and B, are positioned on the x-axis at
    x = 0 m and x = +1.5!m. The loudspeakers act as isotropic
    point sources, operating in phase and emitting pure tones of
    frequency 340 Hz. Each loudspeaker emits a power of
    10.0!W uniformly in all directions. The speed of sound in
    air is 340!m/s, and absorption in the air may be neglected.
    [5] (a) Find the resultant acoustic intensity produced by the
    loudspeakers at point P located on the x-axis at x=+20.0 m.
    Note that interference effects must be considered in
    calculating this intensity, and that 1/r2 attenuation (i.e. geometrical spreading of the sound
    waves) should be taken into account. Assume that there is no blocking of the sound
    transmission from the more distant speaker (A) by the one nearer to P.
    [4] (b) If you started at point P and travelled along the circumference of the circle shown in the
    diagram, what distance (arc length) would you have to travel before you encountered a
    local maximum in the acoustic intensity due to the two coherent sources?
    ?

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    A1.

    Y (x, t) = 0.05 sin (60t 2x 1)

    The standard form of a wave traveling in the +ve x direction is, y (x, t) = A Sin (t - kx + )...(1)

    Where  = 2 pi f and k = 2pi/ where f is the frequency and  is the wavelength

    Comparing, 2  f = 60 which gives f = 60/2*3.14 = 9.55 Hz

    We have freq * wavelength = velocity

    Wavelength  = 2 /k = 3.14 m

    Velocity v = 3.14 * 9.55 = 30 m/s

    The wave is traveling in the -ve x direction, from equation (1)

    The maximum transverse velocity is given by, v = 4A/T = 4A*f = 4*0.05 * 9.55 = 1.91 m/s

    A2

    We have for acoustic impedance, z = density of the medium * velocity of sound

    Here we have two media with different densities d1 and d2 and thus we will have two z's as z1 and z2.

    It is given that the density is proportional to 1/T

    Thus, D1  1/T1 and D2  1/T2

    Thus Z1  (1/T1) V1 where V1 is the velocity in medium 1

    And Z2  (1/T2) V2

    Need not worry about the proportionality constant as we are going to take a ratio,

    Now, Z1/Z2 = [(1/T1) V1] / [(1/T2) V2]

    = (V1/T1)*(T2/V2)

    If the velocities v1 and v2 are the same in both the media (since pressure is almost the same)

    We get Z1/Z2 = T2/T1 with T1 = 273 - 40 = 233 K and T2 = 273 + 20 = 293 K,

    We get, Z1/Z2 = 293/233 = 1.26

    Since the impedance is inversely proportional to temperature, the medium with lower temperature will have higher impedance (in our case, air with -40 degree celcius)

    Accoustic impedance has the unit kg/(m2 S)
    Reflection is often quantified in term of the reflection coefficient 'R'. R is defined simply as the ratio of the reflected and incident wave amplitudes.
    R = ar / ai
    Where 'ai' and 'ar' are the incident and reflected wave amplitudes respectively.
    It is also given by, where alpha I is the incident angle (here =0)
    We are sending a wave from cold air (z1) to warmer (z2). We already have, z1 > z2
    Thus, R will be ...

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