Name________________________

TEST2 SP10

1. A transverse traveling wave on a string is represented by y = Asin(kx – ωt + Φ), where y is in cm, x is in meters, t is in seconds and Φ is a phase constant . The amplitude is A= 2.00 cm, the wavelength λ = 4.00 cm, and the frequency f = 246 Hz.

At t = 0 and x = 0, the string has a transverse displacement y = 1.50 cm. The positive direction of y is vertically up. At that moment (t = 0) and position (x = 0), a particle of the string is moving downward; thus the transverse velocity v_y = < 0.

(a) (6 points) What direction does the wave travel, in the positive or negative x direction. Circle one. Prove your answer as shown in class.

(b) (20 points) What is the phase constant Φ ( in radians)?

(c) (9 points) Carefully sketch the wave as a function of x at t = 0 on the axes shown below for the range shown. Carefully label the y-intercepts and x-intercepts for the range shown .

(d) (7 points) In your sketch, is a particle of the string at x = 0 moving vertically up or vertically down? Evaluate (compute) the transverse velocity of a particle of the string at x = 0 and t = 0. Is your answer positive or negative?

2. In the figure below, water flows out of a small cylindrical tank. Water is flowing out of an open circular valve of radius r at a distance h below the top fluid surface. The radius of the top water surface is 3r. The pressures of the water at the top and at the open valve are equal to atmospheric pressure. The water at the top surface descends at speed
v₁ ; the water at the valve below has speed v₂ .

(a) (20 points) Derive an expression for v₂ in terms of symbols g, h and other numerical constants. Simplify as much as possible.

(b) (8 points) Suppose h = 2.40 m and r = 5.00 cm. How long (in minutes) would it take water from the open valve to fill a pool (not shown) of diameter 6.0 m and depth 1.2 m.

3. One end of a horizontal string of linear mass density 6.6x10 ^-4 kg/m is attached to a small amplitude mechanical oscillator with frequency f = 120 Hz. The string passes over a pulley a distance L = 1.50 m away and weights of mass m are hung from the right end. Suppose m = 2.18 kg. The mechanical oscillator sets up a standing wave pattern along the string described by,
y = 2A·cosωt· sinkx, where A = 2.40 cm.

Here x = 0 at the left end and x = L at the pulley. Both fixed ends are nodes. The positive y direction is considered to be vertically up.
(a) (15 points) How many loops of a standing wave are produced on the string?
(b) (5 points) What is k ?

(c) (5 points) What is ω?
(d) (15 points) Find the transverse velocity v_y of a particle of the string when x = 0.375 m and t = 0.002 seconds.
(e) ( 4 points) In part (d), is the particle of the string moving up or down? Explain.
(f) (10 points) Assume y = y₁ + y₂. If y₁ = Asin(kx + ωt), then what is the expression for y₂? Express your answer in terms of A, k, x, ω, t and other symbols. You must check your answer by adding it to y₁ and showing that the sum equals 2A·cosωt· sinkx. Show all work !
(g) (4 points) Substitute the values given in this problem and compute the speed v (in m/s) of wave y₂ discussed in part (f).

(h) (4 points) Showing the loops, roughly sketch the standing wave on the diagram below at seven times , i.e. with three curves above and three curves below the string axis. Carefully label all nodes.

4. (Extra Credit) ( Tentatively 10 points) A firework shell explodes a distance h = 111 m above the ground, creating a colorful display of sparks and a loud sound. Person A standing at point A directly below the explosion detects a sound level of 110 db. Person B stands a distance D to the right of person A and detects a sound level of 102 dB.

(a) (6 points) What is the distance D?
(b) (4 points) How much sooner does person A hear the sound before person B? Could the two people, using only their ears, distinguish the two arrival times? Explain. Assume the speed of sound is 331 m/s.