Physics: Mod 1: Standing Waves & Beats

Item 1 (a) is a snapshot graph at t = 0 s of two waves approaching each other at 1.0 m/s. At what time was the snapshot graph (b) taken? Express your answer to two significant figures and include the appropriate units.

Item 2 (a) is a snapshot graph at t = 0 s of two waves approaching each other at 1.0 m/s. At what time was the snapshot graph (b) taken? Express your answer to two significant figures and include the appropriate units.

Item 3 (a) is a snapshot graph at t = 0 s of two waves approaching each other at 1.0 m/s. At what time was the snapshot graph (b) taken? Express your answer to two significant figures and include the appropriate units.

Item 4 Consider a traveling wave described by the formula y1(x,t)=Asin(kx−ωt). This function might represent the lateral displacement of a string, a local electric field, the position of the surface of a body of water, or any of a number of other physical manifestations of waves. Part A: Which one of the following statements about the wave described in the problem introduction is correct? Part B: Which of the expressions given is a mathematical expression for a wave of the same amplitude that is traveling in the opposite direction? At time t=0 this new wave should have the same displacement as y1(x,t) the wave described in the problem introduction. Part C: Find ye(x)and yt(t). Keep in mind that yt(t) should be a trigonometric function of unit amplitude.Separate the two functions with a comma. Part D: Which one of the following statements about the superposition wave ys(x,t) is correct? Part E: At the position x=0, what is the displacement of the string (assuming that the standing wave ys(x,t) is present)? Express your answer in terms of parameters given in the problem introduction. Part F: At certain times, the string will be perfectly straight. Find the first time t1 > 0 when this is true. Express t1 in terms of ω, k, and necessary constants. Part G: From Part F we know that the string is perfectly straight at time t=π/2ω. Which of the following statements does the string's being straight imply about the energy stored in the string?

Item 5 yi(x,t)=Ai sin(2π(x/λi)) sin(2π(fi)t) A: The string described in the problem introduction is oscillating in one of its normal modes. Which of the following statements about the wave in the string is correct? B: Which of the following statements are true? C: Find the three longest wavelengths (call them λ1, λ2, and λ3) that "fit" on the string, that is, those that satisfy the boundary conditions at x=0 and x=L. These longest wavelengths have the lowest frequencies. Express the three wavelengths in terms of L. List them in decreasing order of length, separated by commas. D: The frequency of each normal mode depends on the spatial part of the wave function, which is characterized by its wavelength λi. Find the frequency fi of the ith normal mode. Express fi in terms of its particular wavelength λi and the speed of propagation of the wave v. E:Find the three lowest normal mode frequencies f1, f2, and f3. Express the frequencies in terms of L, v, and any constants. List them in increasing order, separated by commas.

Item 6 A combination work of art/musical instrument is illustrated. (Figure 1) Six pieces of identical piano wire (cut to different lengths) are hung from the same support, and masses are hung from the free end of each wire. Each wire is 1, 2, or 3 units long, and each supports 1, 2, or 4 units of mass. The mass of each wire is negligible compared to the total mass hanging from it. When a strong breeze blows, the wires vibrate and create an eerie sound. A: Rank each wire-mass system on the basis of its fundamental wavelength. Rank from largest to smallest. To rank items as equivalent, overlap them. B: Rank each wire-mass system on the basis of its wave speed. Rank from largest to smallest. To rank items as equivalent, overlap them. (Use Wire length to find answer). C: Rank each wire-mass system on the basis of its fundamental frequency. Rank from largest to smallest.

Item 7 (Figure 1) shows a standing wave oscillating at 100 Hz on a string. 6 nodes and 7 anti-nodes. The string is 50cm. What is the wave speed?

Item 8 (Figure 1) shows a standing wave on a 1.9 m long string that has been fixed at both ends and tightened until the wave speed is 50 m/s. 3 nodes and 4 anti-nodes What is the frequency?