91Ó°ÊÓ

The equation of a transverse wave traveling along a very long string is $\mathit{y}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{0}\mathit{s}\mathit{i}\mathit{n}\left(0.020\mathrm{Ï€³\ae }+4.0\mathrm{Ï€³Ù}\right)$, where x andy are expressed in centimeters and is in seconds. (a) Determine the amplitude,(b) Determine the wavelength, (c)Determine the frequency, (d) Determine the speed, (e) Determine the direction of propagation of the wave, and (f) Determine the maximum transverse speed of a particle in the string. (g)What is the transverse displacement atx = 3.5 cmwhen t = 0.26 s?

If you set up the seventh harmonic on a string, (a) how many nodes are present, and (b) is there a node, antinode, or some intermediate state at the midpoint? If you next set up the sixth harmonic, (c) is its resonant wavelength longer or shorter than that for the seventh harmonic, and (d) is the resonant frequency higher or lower?

A sinusoidal transverse wave of wavelength 20cmtravels along a string in the positive direction of anaxis. The displacement y of the string particle at x=0is given in Figure 16-34 as a function of time t. The scale of the vertical axis is set by${\mathit{y}}_{\mathbf{s}}\mathbf{=}\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{cm}$The wave equation is to be in the form$\mathbf{y}\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{t}\mathbf{)}\mathbf{=}{\mathbf{y}}_{\mathbf{m}}\mathbf{sin}\mathbf{(}\mathbf{kx}\mathbf{Â\pm }\mathbf{Ó\neg ³Ù}\mathbf{+}\mathbf{Ï•}\mathbf{)}$. (a) At t=0, is a plot of y versus x in the shape of a positive sine function or a negative sine function? (b) What is${\mathit{y}}_{\mathbf{m}}$, (c) What isk,(d) What is$\mathit{Ó\neg }$, (e) What is$\mathit{φ}$ (f) What is the sign in front of$\mathit{Ó\neg }$, and (g) What is the speed of the wave? (h) What is the transverse velocity of the particle at x=0when t=5.0 s?

The function$\mathbf{y}\left(x,t\right)\mathbf{=}\left(15.0\mathrm{cm}\right)\mathbf{}\mathbf{cos}\mathbf{}\left(\mathrm{Ï„Ï„³\ae }-15\mathrm{Ï„Ï„³Ù}\right)$, with x in meters and t in seconds, describes a wave on a taut string. What is the transverse speed for a point on the string at an instant when that point has the displacement $\mathit{y}\mathbf{=}\mathbf{+}\mathbf{12}\mathbf{.}\mathbf{0}\mathbf{}\mathit{c}\mathit{m}$?

A sinusoidal wave of frequency 500 Hzhas a speed of 350 m/s . (a)How far apart are two points that differ in phase by$\mathbf{Ï€}\mathbf{/}\mathbf{3}\mathbf{rad}$ ? (b)What is the phase difference between two displacements at a certain point at times 1.00 msapart?

The equation of a transverse wave on a string is$\mathit{y}\mathbf{=}\left(2.0mm\right)\mathit{s}\mathit{i}\mathit{n}\left[\left(20m^{-1}\right)x-\left(600s^{-1}\right)t\right]$ . The tension in the string is 15 N . (a)What is the wave speed? (b)Find the linear density of this string in grams per meter.

The speed of a transverse wave on a string is170m/swhen the string tension is 120N. To what value must the tension be changed to raise the wave speed to180m/s?

The linear density of a string is$\mathbf{1}\mathbf{.}\mathbf{6}\mathbf{×}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}\mathit{k}\mathit{g}\mathbf{/}\mathit{m}$. A transverse wave on the string is described by the equation$\mathit{y}\mathbf{=}\left(0.021m\right)\mathbf{}\mathit{s}\mathit{i}\mathit{n}\left[\left(2.0m^{-1}\right)x+\left(30s^{-1}\right)t\right]$. (a)What are the wave speed and (b) What is the tension in the string?

If a wave$\mathit{y}\left(x,t\right)\mathbf{=}\left(6.0mm\right)\mathbf{}\mathit{s}\mathit{i}\mathit{n}\left(kx+\left(600rad/s\right)t\right)$travels along a string, how much time does any given point on the string take to move between displacements$\mathit{y}\mathbf{=}\mathbf{+}\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{}\mathit{m}\mathit{m}$and $\mathit{y}\mathbf{=}\mathbf{-}\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{}\mathit{m}\mathit{m}$?

The following four waves are sent along strings with the same linear densities (xis in meters and tis in seconds). Rank the waves according to (a) their wave speed and (b) the tension in the strings along which they travel, greatest first:

(1)${\mathit{Y}}^{\mathbf{1}}\mathbf{=}\mathbf{\left(}\mathbf{3}\mathit{m}\mathit{m}\mathbf{\right)}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathit{x}\mathbf{-}\mathbf{3}\mathit{t}\mathbf{)}\mathbf{,}$ (3)${\mathit{y}}^{\mathbf{3}}\mathbf{=}\mathbf{\left(}\mathbf{1}\mathit{m}\mathit{m}\mathbf{\right)}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathbf{4}\mathit{x}\mathbf{-}\mathit{t}\mathbf{)}$,

(2) ${\mathit{y}}^{\mathbf{2}}\mathbf{=}\mathbf{\left(}\mathbf{6}\mathit{m}\mathit{m}\mathbf{\right)}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathbf{2}\mathit{x}\mathbf{-}\mathit{t}\mathbf{)}$, (4)${\mathit{y}}^{\mathbf{4}}\mathbf{=}\mathbf{\left(}\mathbf{2}\mathit{m}\mathit{m}\mathbf{\right)}\mathbf{}\mathit{s}\mathit{i}\mathit{n}\mathbf{(}\mathit{x}\mathbf{-}\mathbf{2}\mathit{t}\mathbf{)}$.