# hw3sol - HW 2 Physics 1B Spring 2010 C M Cooper Teaching...

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HW 2: Physics 1B Spring 2010 C. M. Cooper, Teaching Assitant W. Gekelman, Professor April 16, 2010 Abstract This hw covers all of chapter 15. Topics include standing waves in pipes and then a ton of random things. These include sound wave velocity, pressure, intensity, interference, beats, and doppler shifts. You should know the speed of sound in air at 20 C = 344 m/s. 1 Question 16.10: speed of sound with temperature This problem involves taking ”a diﬀerential” then using it in a taylor series. These sorts of things are goign to become more commonplace as you move forward. Starting at equation 16.10 v = r γRT M (1) dv = d ± r γRT M ² = r γR M dT 2 T = r γR M dT 1 2 T T T = v dT 2 T (2) now we use a ”ﬁrst order taylor expansion” which is the same as splitting the Δ v into v f - v i Δ v = dv dT Δ T = v 2 T Δ T = 344 m/s 2 × 293 C (1 C ) = 0 . 59 m/s (3) 2 Question 16.25: Waves in Tubes This problem is best done graphically and writing out the answer from the pictures. The most important part of this problem is knowing the diﬀerence between the pressure nodes and the am- plitude nodes, and that they are π/ 2 out of phase (pressure nodes occur at amplitude antinodes and vice versa). You can do this with formulas for λ n if you know them but you still need to draw the pics, here they are below. For the fundamental, the displacement node is at 0.6 m and the pressure nodes (displacement antinodes) are at 0 m and 1.20 m. For the ﬁrst overtone (second harmonic), the displacement nodes are at 0.3 m and 0.9 m and the pressure nodes are at 0 m, 0.6 m and 1.2 m. For the second overtone (third harmonic), the displacement nodes are at 0.2 m 0.6 m and and 1.0 m and the pressure nodes are at 0 m, 0.4 m, 0.8 m and 1.2 m. 1

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Figure 1: This is best done graphically Figure 2: hey, hey, hey. goodbye. For the fundamental, the displacement node is at 0 m and the pressure node (displacement antinodes) is at 1.20 m. For the ﬁrst overtone (second harmonic), the displacement nodes are at 0.0 m and 0.8 m and the pressure nodes are at 0.4 m and 1.2 m. For the second overtone (third harmonic), the displacement nodes are at 0.0 m 0.48 m and and 0.96 m and the pressure nodes are at 0.24 m, 0.72 m, and 1.2 m. 3 Question 16.32: a Can on a String You’re going to be making a wave in one medium and converting it to another. Remember when you see μ or F T think v which if you have λ you can get anything k,ω,f . The question is though, what do you do at the boundary? You might be tempted to match the velocities but what you actually want is for them to resonate together, i.e. they have the same frequencies. This means f pipe = f wire . Subbing in for the n th frequency of both gives v s 4 L pipe = 3 v wire 2 L wire = 3 2 L wire s F m/L wire (4) rearranging gives L pipe = L wire v s 6 p FL wire /m = . 85 × 344 6 p 4110 × . 85 /. 00725 = 0 . 07 m (5) 2
4 Question 16.43: How’d That Guy get There?

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hw3sol - HW 2 Physics 1B Spring 2010 C M Cooper Teaching...

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