1507 Standing Waves On a String

Exercises

36, 38, 40, 42

15.36 Adjacent antinodes of a standing wave on a string are 15.0 cm15.0 \text{ cm} apart. A particle at an antinode oscillates in simple harmonic motion with amplitude 0.850 cm0.850 \text{ cm} and period 0.0750 s0.0750 \text{ s}. The string lies along the +x-axis and is fixed at x=0x = 0.
(a) How far apart are the adjacent nodes?
(b) What are the wavelength, amplitude, and speed of the two traveling waves that form this pattern?
(c) Find the maximum and minimum transverse speeds of a point at an antinode.
(d) What is the shortest distance along the string between a node and an antinode?

Solution

a. Thd distance between two adjacent nodes is 15.0 cm15.0 \text{ cm}.
b. Wavelength λ=0.15×2=0.3 m\lambda = 0.15 \times 2 = 0.3 \text{ m}, amplitude A=0.85×10−2 mA = 0.85 \times 10^{-2} \text{ m}
c. Todo
d. The shortest distance between a node and an antinode is 0.152=0.075 m\frac{0.15}{2} = 0.075 \text{ m}.

15.38 A 1.501.50-m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 62.0 m/s62.0 \text{ m/s}. What are the wavelength and frequency of (a) the fundamental; (b) the second overtone; (c) the fourth harmonic?

Solution

a. Fundamental

n=1,λ=2Ln=2×1.51=3 mf=vλ=623=20.67 Hz\begin{aligned} n &= 1, \lambda = \frac{2L}{n} = \frac{2\times 1.5}{1} = 3\text{ m}\\ f &= \frac{v}{\lambda} = \frac{62}{3} = 20.67 \text{ Hz} \end{aligned}

b. The second overtone

n=3,λ=2L3=2×1.53=1 mf=vλ=621=60 Hz\begin{aligned} n &= 3, \lambda = \frac{2L}{3} = \frac{2\times 1.5}{3} = 1\text{ m}\\ f &= \frac{v}{\lambda} = \frac{62}{1} = 60 \text{ Hz} \end{aligned}

c. The fourth overtone

n=4,λ=2L4=2×1.54=0.75 mf=vλ=620.75=82.67 Hz\begin{aligned} n &= 4, \lambda = \frac{2L}{4} = \frac{2\times 1.5}{4} = 0.75\text{ m}\\ f &= \frac{v}{\lambda} = \frac{62}{0.75} = 82.67 \text{ Hz} \end{aligned}

15.40 A piano tuner stretches a steel piano wire with a tension of 800N800 N. The steel wire is 0.400m0.400 m long and has a mass of 3.00g3.00 g. (a) What is the frequency of its fundamental mode of vibration? (b) What is the number of the highest harmonic that could be heard by a person who is capable of hearing frequencies up to 10,000 Hz10,000 \text{ Hz}?

Solution

Todo

15.42 The wave function of a standing wave is y(x,t)=4.44 mmsin⁡[(32.5 rad/m)xsin⁡[(754 rad/s)t]y(x, t) = 4.44 \text{ mm} \sin[(32.5 \text{ rad/m}) x \sin[(754 \text{ rad/s}) t]. For the two traveling waves that make up this standing wave, find the (a) amplitude; (b) wavelength; (c) frequency; (d) wave speed; (e) wave functions. (f) From the information given, can you determine which harmonic this is? Explain.

Solution

Todo