# 16.6 Standing waves and resonance  (Page 7/17)

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Why do roofs of gymnasiums and churches seem to fail more than family homes when an earthquake occurs?

Wine glasses can be set into resonance by moistening your finger and rubbing it around the rim of the glass. Why?

Energy is supplied to the glass by the work done by the force of your finger on the glass. When supplied at the right frequency, standing waves form. The glass resonates and the vibrations produce sound.

Air conditioning units are sometimes placed on the roof of homes in the city. Occasionally, the air conditioners cause an undesirable hum throughout the upper floors of the homes. Why does this happen? What can be done to reduce the hum?

Consider a standing wave modeled as $y\left(x,t\right)=4.00\phantom{\rule{0.2em}{0ex}}\text{cm}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(3\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x\right)\text{cos}\left(4\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t\right).$ Is there a node or an antinode at $x=0.00\phantom{\rule{0.2em}{0ex}}\text{m}?$ What about a standing wave modeled as $y\left(x,t\right)=4.00\phantom{\rule{0.2em}{0ex}}\text{cm}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(3\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x+\frac{\pi }{2}\right)\text{cos}\left(4\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t\right)?$ Is there a node or an antinode at the $x=0.00\phantom{\rule{0.2em}{0ex}}\text{m}$ position?

For the equation $y\left(x,t\right)=4.00\phantom{\rule{0.2em}{0ex}}\text{cm}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(3\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x\right)\text{cos}\left(4\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t\right),$ there is a node because when $x=0.00\phantom{\rule{0.2em}{0ex}}\text{m}$ , $\text{sin}\left(3\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}\left(0.00\phantom{\rule{0.2em}{0ex}}\text{m}\right)\right)=0.00,$ so $y\left(0.00\phantom{\rule{0.2em}{0ex}}\text{m},t\right)=0.00\phantom{\rule{0.2em}{0ex}}\text{m}$ for all time. For the equation $y\left(x,t\right)=4.00\phantom{\rule{0.2em}{0ex}}\text{cm}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(3\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x+\frac{\pi }{2}\right)\text{cos}\left(4\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t\right),$ there is an antinode because when $x=0.00\phantom{\rule{0.2em}{0ex}}\text{m}$ , $\text{sin}\left(3\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}\left(0.00\phantom{\rule{0.2em}{0ex}}\text{m}\right)+\frac{\pi }{2}\right)=+1.00$ , so $y\left(0.00\phantom{\rule{0.2em}{0ex}}\text{m},t\right)$ oscillates between + A and − A as the cosine term oscillates between +1 and -1.

## Problems

A wave traveling on a Slinky® that is stretched to 4 m takes 2.4 s to travel the length of the Slinky and back again. (a) What is the speed of the wave? (b) Using the same Slinky stretched to the same length, a standing wave is created which consists of three antinodes and four nodes. At what frequency must the Slinky be oscillating?

A 2-m long string is stretched between two supports with a tension that produces a wave speed equal to ${v}_{w}=50.00\phantom{\rule{0.2em}{0ex}}\text{m/s}.$ What are the wavelength and frequency of the first three modes that resonate on the string?

$\begin{array}{}\\ \\ {\lambda }_{n}=\frac{2.00}{n}L,\phantom{\rule{1em}{0ex}}{f}_{n}=\frac{v}{{\lambda }_{n}}\hfill \\ {\lambda }_{1}=4.00\phantom{\rule{0.2em}{0ex}}\text{m},\phantom{\rule{1em}{0ex}}{f}_{1}=12.5\phantom{\rule{0.2em}{0ex}}\text{Hz}\hfill \\ {\lambda }_{2}=2.00\phantom{\rule{0.2em}{0ex}}\text{m},\phantom{\rule{1em}{0ex}}{f}_{2}=25.00\phantom{\rule{0.2em}{0ex}}\text{Hz}\hfill \\ {\lambda }_{3}=1.33\phantom{\rule{0.2em}{0ex}}\text{m},\phantom{\rule{1em}{0ex}}{f}_{3}=37.59\phantom{\rule{0.2em}{0ex}}\text{Hz}\hfill \end{array}$

Consider the experimental setup shown below. The length of the string between the string vibrator and the pulley is $L=1.00\phantom{\rule{0.2em}{0ex}}\text{m}.$ The linear density of the string is $\mu =0.006\phantom{\rule{0.2em}{0ex}}\text{kg/m}.$ The string vibrator can oscillate at any frequency. The hanging mass is 2.00 kg. (a)What are the wavelength and frequency of $n=6$ mode? (b) The string oscillates the air around the string. What is the wavelength of the sound if the speed of the sound is ${v}_{s}=343.00\phantom{\rule{0.2em}{0ex}}\text{m/s?}$ A cable with a linear density of $\mu =0.2\phantom{\rule{0.2em}{0ex}}\text{kg/m}$ is hung from telephone poles. The tension in the cable is 500.00 N. The distance between poles is 20 meters. The wind blows across the line, causing the cable resonate. A standing waves pattern is produced that has 4.5 wavelengths between the two poles. The air temperature is $T=20\text{°}\text{C}.$ What are the frequency and wavelength of the hum?

$\begin{array}{}\\ \\ v=158.11\phantom{\rule{0.2em}{0ex}}\text{m/s,}\phantom{\rule{1em}{0ex}}\lambda =4.44\phantom{\rule{0.2em}{0ex}}\text{m,}\phantom{\rule{1em}{0ex}}f=35.61\phantom{\rule{0.2em}{0ex}}\text{Hz}\hfill \\ {\lambda }_{s}=9.63\phantom{\rule{0.2em}{0ex}}\text{m}\hfill \end{array}$

Consider a rod of length L , mounted in the center to a support. A node must exist where the rod is mounted on a support, as shown below. Draw the first two normal modes of the rod as it is driven into resonance. Label the wavelength and the frequency required to drive the rod into resonance. Consider two wave functions $y\left(x,t\right)=0.30\phantom{\rule{0.2em}{0ex}}\text{cm}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(3\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x-4\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t\right)$ and $y\left(x,t\right)=0.30\phantom{\rule{0.2em}{0ex}}\text{cm}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(3\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x+4\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t\right)$ . Write a wave function for the resulting standing wave.

$y\left(x,t\right)=\left[0.60\phantom{\rule{0.2em}{0ex}}\text{cm}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(3\phantom{\rule{0.2em}{0ex}}{\text{m}}^{-1}x\right)\right]\text{cos}\left(4\phantom{\rule{0.2em}{0ex}}{\text{s}}^{-1}t\right)$

#### Questions & Answers

a particle projected from origin moving on x-y plane passes through P & Q having consituents (9,7) , (18,4) respectively.find eq. of trajectry.
rahul Reply
definition of inertia
philip Reply
the reluctance of a body to start moving when it is at rest and to stop moving when it is in motion
charles
An inherent property by virtue of which the body remains in its pure state or initial state
Kushal
why current is not a vector quantity , whereas it have magnitude as well as direction.
Aniket Reply
why
daniel
the flow of current is not current
fitzgerald
bcoz it doesn't satisfy the algabric laws of vectors
Shiekh
The Electric current can be defined as the dot product of the current density and the differential cross-sectional area vector : ... So the electric current is a scalar quantity . Scalars are related to tensors by the fact that a scalar is a tensor of order or rank zero .
Kushal
what is binomial theorem
Tollum Reply
hello are you ready to ask aquestion?
Saadaq Reply
what is binary operations
Tollum
What is the formula to calculat parallel forces that acts in opposite direction?
Martan Reply
position, velocity and acceleration of vector
Manuel Reply
hi
peter
hi
daniel
hi
Vedisha
*a plane flies with a velocity of 1000km/hr in a direction North60degree east.find it effective velocity in the easterly and northerly direction.*
imam
hello
Lydia
hello Lydia.
Sackson
What is momentum
isijola
hello
Saadaq
A rail way truck of mass 2400kg is hung onto a stationary trunk on a level track and collides with it at 4.7m|s. After collision the two trunk move together with a common speed of 1.2m|s. Calculate the mass of the stationary trunk
Ekuri Reply
I need the solving for this question
philip
is the eye the same like the camera
EDWIN Reply
I can't understand
Suraia
same here please
Josh
I think the question is that ,,, the working principal of eye and camera same or not?
Sardar
yes i think is same as the camera
muhammad
what are the dimensions of surface tension
samsfavor
why is the "_" sign used for a wave to the right instead of to the left?
MUNGWA Reply
why classical mechanics is necessary for graduate students?
khyam Reply
classical mechanics?
Victor
principle of superposition?
Naveen Reply
principle of superposition allows us to find the electric field on a charge by finding the x and y components
Kidus
Two Masses,m and 2m,approach each along a path at right angles to each other .After collision,they stick together and move off at 2m/s at angle 37° to the original direction of the mass m. What where the initial speeds of the two particles
MB
2m & m initial velocity 1.8m/s & 4.8m/s respectively,apply conservation of linear momentum in two perpendicular directions.
Shubhrant
A body on circular orbit makes an angular displacement given by teta(t)=2(t)+5(t)+5.if time t is in seconds calculate the angular velocity at t=2s
MB
2+5+0=7sec differentiate above equation w.r.t time, as angular velocity is rate of change of angular displacement.
Shubhrant
Ok i got a question I'm not asking how gravity works. I would like to know why gravity works. like why is gravity the way it is. What is the true nature of gravity?
Daniel Reply
gravity pulls towards a mass...like every object is pulled towards earth
Ashok
An automobile traveling with an initial velocity of 25m/s is accelerated to 35m/s in 6s,the wheel of the automobile is 80cm in diameter. find * The angular acceleration
Goodness Reply
(10/6) ÷0.4=4.167 per sec
Shubhrant
what is the formula for pressure?
Goodness Reply
force/area
Kidus
force is newtom
Kidus
and area is meter squared
Kidus
so in SI units pressure is N/m^2
Kidus
In customary United States units pressure is lb/in^2. pound per square inch
Kidus

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