Chapter 2: Q41E (page 500)

A thin, taut string tied at both ends and oscillating in its third harmonic has its shape described by the equation \(y\left( {x,t} \right) = \left( {5.6\;{\rm{cm}}} \right)\sin \left[ {\left( {0.0340\;{{{\rm{rad}}} \mathord{\left/ {\vphantom {{{\rm{rad}}} {{\rm{cm}}}}} \right. \\} {{\rm{cm}}}}} \right)x} \right]\sin \left[ {\left( {50.0\;{{{\rm{rad}}} \mathord{\left/ {\vphantom {{{\rm{rad}}} {\rm{s}}}} \right. \\} {\rm{s}}}} \right)t} \right]\), where the origin is at the left end of the string, the x-axis is along the string, and the y-axis is perpendicular to the string.
(a) Draw a sketch that shows the standing-wave pattern.
(b) Find the amplitude of the two traveling waves that make up this standing wave.
(c) What is the length of the string?
(d) Find the wavelength, frequency, period, and speed of the traveling waves.
(e) Find the maximum transverse speed of a point on the string.
(f) What would be the equation \(y\left( {x,t} \right)\)for this string if it were vibrating in its eighth harmonic?

Short Answer

Expert verified

(a) The sketch is attached below.

Step by step solution

Identification of the given data

The given data can be listed below as,

The equation is, \(y\left( {x,t} \right) = \left( {5.6\;{\rm{cm}}} \right)\sin \left[ {\left( {0.0340\;{{{\rm{rad}}} \mathord{\left/ {\vphantom {{{\rm{rad}}} {{\rm{cm}}}}} \right. \\} {{\rm{cm}}}}} \right)x} \right]\sin \left[ {\left( {50.0\;{{{\rm{rad}}} \mathord{\left/ {\vphantom {{{\rm{rad}}} {\rm{s}}}} \right. \\} {\rm{s}}}} \right)t} \right]\).

Significance of the standing wave pattern

The lowest energy vibrational modes of the object are represented by these standing wave patterns. Although an object can vibrate in innumerable ways (each associated with a different frequency), most things prefer to vibrate in a small number of distinct ways.