91Ó°ÊÓ

Problem 50

Tension is maintained in a string by attaching one end to a wall and by hanging a \(2.20-\mathrm{kg}\) object from the other end of the string after it passes over a pulley that is \(2.00 \mathrm{m}\) from the wall. The string has a mass per unit length of \(3.55 \mathrm{mg} / \mathrm{m} .\) What is the fundamental frequency of this string?

Problem 51

A guitar's E-string has length \(65 \mathrm{cm}\) and is stretched to a tension of \(82 \mathrm{N}\). It vibrates at a fundamental frequency of $329.63 \mathrm{Hz}$ Determine the mass per unit length of the string.

Problem 52

A string \(2.0 \mathrm{m}\) long is held fixed at both ends. If a sharp blow is applied to the string at its center, it takes \(0.050 \mathrm{s}\) for the pulse to travel to the ends of the string and return to the middle. What is the fundamental frequency of oscillation for this string?

Problem 53

A 1.6 -m-long string fixed at both ends vibrates at resonant frequencies of \(780 \mathrm{Hz}\) and \(1040 \mathrm{Hz}\), with no other resonant frequency between these values. (a) What is the fundamental frequency of this string? (b) When the tension in the string is \(1200 \mathrm{N},\) what is the total mass of the string?

Problem 55

The longest "string" (a thick metal wire) on a particular piano is $2.0 \mathrm{m}\( long and has a tension of \)300.0 \mathrm{N} .$ It vibrates with a fundamental frequency of \(27.5 \mathrm{Hz}\). What is the total mass of the wire?

Problem 56

Suppose that a string of length \(L\) and mass \(m\) is under tension \(F\). (a) Show that \(\sqrt{F L} m\) has units of speed. (b) Show that there is no other combination of \(L, m,\) and \(F\) with units of speed. [Hint: Of the dimensions of the three quantities $L, m, \text { and } F, \text { only } F \text { includes time. }]$ Thus, the speed of transverse waves on the string can only be some dimensionless constant times \(\sqrt{F L / m}.\)

Problem 57

Comprehensive Problems The speed of waves on a lake depends on frequency. For waves of frequency \(1.0 \mathrm{Hz}\), the wave speed is \(1.56 \mathrm{m} / \mathrm{s} ;\) for \(2.0-\mathrm{Hz}\) waves, the speed is \(0.78 \mathrm{m} / \mathrm{s} .\) The 2.0-Hz waves from a speedboat's wake reach you \(120 \mathrm{s}\) after the 1.0 -Hz waves generated by the same boat. How far away is the boat?

Problem 58

A transverse wave on a string is described by $y(x, t)=(1.2 \mathrm{cm}) \sin [(0.50 \pi \mathrm{rad} / \mathrm{s}) t-(1.00 \pi \mathrm{rad} / \mathrm{m}) x]$ Find the maximum velocity and the maximum acceleration of a point on the string. Plot graphs for displacement \(y\) versus \(t\), velocity \(v_{y}\) versus \(t\), and acceleration \(a_{y}\) versus \(t\) at \(x=0.\)

Problem 59

What is the wavelength of the radio waves transmitted by an FM station at 90 MHz? (Radio waves travel at \(3.0 \times 10^{8} \mathrm{m} / \mathrm{s} .\)

Problem 60

A longitudinal wave has a wavelength of \(10 \mathrm{cm}\) and an amplitude of \(5.0 \mathrm{cm}\) and travels in the \(y\) -direction. The wave speed in this medium is \(80 \mathrm{cm} / \mathrm{s},\) (a) Describe the motion of a particle in the medium as the wave travels through the medium. (b) How would your answer differ if the wave were transverse instead?