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The given graph is of amplitude, X, along y axis versus driving frequency f, along x axis. There are three points on the x axis as f not divided by two, f not, three multiply f not divided by two. There are three curves along the x axis, in a one crest oscillation way, which are one over each other in correspondence. The curves start at a point just over the origin point and ends up at a same level along the x axis on the far right. The crests of the three curves are exactly over the f not point. The uppermost crest shows the small damping, whereas the middle one shows the medium damping, and the last one below shows the heavy damping.
Amplitude of a harmonic oscillator as a function of the frequency of the driving force. The curves represent the same oscillator with the same natural frequency but with different amounts of damping. Resonance occurs when the driving frequency equals the natural frequency, and the greatest response is for the least amount of damping. The narrowest response is also for the least damping.

It is interesting that the widths of the resonance curves shown in [link] depend on damping: the less the damping, the narrower the resonance. The message is that if you want a driven oscillator to resonate at a very specific frequency, you need as little damping as possible. Little damping is the case for piano strings and many other musical instruments. Conversely, if you want small-amplitude oscillations, such as in a car’s suspension system, then you want heavy damping. Heavy damping reduces the amplitude, but the tradeoff is that the system responds at more frequencies.

These features of driven harmonic oscillators apply to a huge variety of systems. When you tune a radio, for example, you are adjusting its resonant frequency so that it only oscillates to the desired station’s broadcast (driving) frequency. The more selective the radio is in discriminating between stations, the smaller its damping. Magnetic resonance imaging (MRI) is a widely used medical diagnostic tool in which atomic nuclei (mostly hydrogen nuclei) are made to resonate by incoming radio waves (on the order of 100 MHz). A child on a swing is driven by a parent at the swing’s natural frequency to achieve maximum amplitude. In all of these cases, the efficiency of energy transfer from the driving force into the oscillator is best at resonance. Speed bumps and gravel roads prove that even a car’s suspension system is not immune to resonance. In spite of finely engineered shock absorbers, which ordinarily convert mechanical energy to thermal energy almost as fast as it comes in, speed bumps still cause a large-amplitude oscillation. On gravel roads that are corrugated, you may have noticed that if you travel at the “wrong” speed, the bumps are very noticeable whereas at other speeds you may hardly feel the bumps at all. [link] shows a photograph of a famous example (the Tacoma Narrows Bridge) of the destructive effects of a driven harmonic oscillation. The Millennium Bridge in London was closed for a short period of time for the same reason while inspections were carried out.

In our bodies, the chest cavity is a clear example of a system at resonance. The diaphragm and chest wall drive the oscillations of the chest cavity which result in the lungs inflating and deflating. The system is critically damped and the muscular diaphragm oscillates at the resonant value for the system, making it highly efficient.

The figure shows a black and white photo of the Tacoma Narrows Bridge, from the left side view. The middle of the bridge is shown here in an oscillating state due to heavy cross winds.
In 1940, the Tacoma Narrows Bridge in Washington state collapsed. Heavy cross winds drove the bridge into oscillations at its resonant frequency. Damping decreased when support cables broke loose and started to slip over the towers, allowing increasingly greater amplitudes until the structure failed (credit: PRI's Studio 360 , via Flickr)

A famous magic trick involves a performer singing a note toward a crystal glass until the glass shatters. Explain why the trick works in terms of resonance and natural frequency.

The performer must be singing a note that corresponds to the natural frequency of the glass. As the sound wave is directed at the glass, the glass responds by resonating at the same frequency as the sound wave. With enough energy introduced into the system, the glass begins to vibrate and eventually shatters.

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Section summary

  • A system’s natural frequency is the frequency at which the system will oscillate if not affected by driving or damping forces.
  • A periodic force driving a harmonic oscillator at its natural frequency produces resonance. The system is said to resonate.
  • The less damping a system has, the higher the amplitude of the forced oscillations near resonance. The more damping a system has, the broader response it has to varying driving frequencies.

Conceptual questions

Why are soldiers in general ordered to “route step” (walk out of step) across a bridge?

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Problems&Exercises

How much energy must the shock absorbers of a 1200-kg car dissipate in order to damp a bounce that initially has a velocity of 0.800 m/s at the equilibrium position? Assume the car returns to its original vertical position.

384 J

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If a car has a suspension system with a force constant of 5 . 00 × 10 4 N/m size 12{5 "." "00" times "10" rSup { size 8{4} } "N/m"} {} , how much energy must the car’s shocks remove to dampen an oscillation starting with a maximum displacement of 0.0750 m?

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(a) How much will a spring that has a force constant of 40.0 N/m be stretched by an object with a mass of 0.500 kg when hung motionless from the spring? (b) Calculate the decrease in gravitational potential energy of the 0.500-kg object when it descends this distance. (c) Part of this gravitational energy goes into the spring. Calculate the energy stored in the spring by this stretch, and compare it with the gravitational potential energy. Explain where the rest of the energy might go.

(a). 0.123 m

(b). −0.600 J

(c). 0.300 J. The rest of the energy may go into heat caused by friction and other damping forces.

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Suppose you have a 0.750-kg object on a horizontal surface connected to a spring that has a force constant of 150 N/m. There is simple friction between the object and surface with a static coefficient of friction μ s = 0 . 100 size 12{μ rSub { size 8{s} } =0 "." "100"} {} . (a) How far can the spring be stretched without moving the mass? (b) If the object is set into oscillation with an amplitude twice the distance found in part (a), and the kinetic coefficient of friction is μ k = 0 . 0850 size 12{μ rSub { size 8{k} } =0 "." "0850"} {} , what total distance does it travel before stopping? Assume it starts at the maximum amplitude.

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Engineering Application: A suspension bridge oscillates with an effective force constant of 1 . 00 × 10 8 N/m . (a) How much energy is needed to make it oscillate with an amplitude of 0.100 m? (b) If soldiers march across the bridge with a cadence equal to the bridge’s natural frequency and impart 1 . 00 × 10 4 J size 12{1 "." "00" times "10" rSup { size 8{4} } "J"} {} of energy each second, how long does it take for the bridge’s oscillations to go from 0.100 m to 0.500 m amplitude?

(a) 5 . 00 × 10 5 J size 12{ {underline {5 "." "00" times "10" rSup { size 8{5} } " J"}} } {}

(b) 1.20 × 10 3 size 12{ {underline {"20" "." "0 mm"}} } {} s

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Questions & Answers

what is temperature
Adeleye Reply
temperature is the measure of degree of hotness or coldness of a body. measured in kelvin
Ahmad
a characteristic which tells hotness or coldness of a body
babar
Average kinetic energy of an object
Kym
average kinetic energy of the particles in an object
Kym
A measure of the quantity of heat contained in an object which arises from the average kinetic energy of the constituent particles of that object. It can be measured using thermometers. It has a unit of kelvin in the thermodynamic scale and degree Celsius in the Celsius scale.
ibrahim
Mass of air bubble in material medium is negative. why?
Hrithik Reply
a car move 6m. what is the acceleration?
Umaru Reply
depends how long
Peter
What is the simplest explanation on the difference of principle, law and a theory
Kym Reply
how did the value of gravitational constant came give me the explanation
Varun Reply
how did the value of gravitational constant 6.67×10°-11Nm2kg-2
Varun
A steel ball is dropped onto a hard floor from a height of 1.50 m and rebounds to a height of 1.45 m. (a) Calculate its velocity just before it strikes the floor.
Kris Reply
9.8m/s?
Muhammad
Sqrt(2*1.5m*9.81m/s^2)
Richard
0.5m* mate.
Muhammad
0.05 I meant.
Muhammad
Guess your solution is correct considering the ball fall from 1.5m height initially.
Muhammad
Sqrt(2*1.5m*9.81m/s^2)
Deepak
How can we compare different combinations of capacitors?
Prakash Reply
find the dimension of acceleration if it's unit is ms-2
Happiness Reply
lt^-2
Ahmad
b=-2 ,a =1
Ahmad
M^0 L^1T^-2
Sneha
what is botany
Masha
it is a branch of science which deal with the study of plants animals and environment
Varun
what is work
Sunday Reply
a boy moving with an initial velocity of 2m\s and finally canes to rest with a velocity of 3m\s square at times 10se calculate it acceleration
Sunday
.
Abdul
6.6 lol 😁😁
Abdul
show ur work
Sunday
sorry..the answer is -10
Abdul
your question is wrong
Abdul
If the boy is coming to rest then how the hell will his final velocity be 3 it'll be zero
Abdul
re-write the question
Nicolas
men i -10 isn't correct.
Stephen
using v=u + at
Stephen
1/10
Happy
ya..1/10 is very correct..
Stephen
hnn
Happy
how did the value 6.67×10°-11Nm2kg2 came tell me please
Varun
Work is the product of force and distance
Kym
physicist
Michael
what is longitudinal wave
Badmus Reply
A longitudinal wave is wave which moves parallel or along the direction of propagation.
sahil
longitudinal wave in liquid is square root of bulk of modulus by density of liquid
harishree
Is British mathematical units the same as the United States units?(like inches, cm, ext.)
Nina Reply
We use SI units: kg, m etc but the US sometimes refer to inches etc as British units even though we no longer use them.
Richard
Thanks, just what I needed to know.
Nina
What is the advantage of a diffraction grating over a double slit in dispersing light into a spectrum?
Uditha Reply
can I ask questions?
Boniface Reply
yes.
Abdul
Yes
Albert
sure
Ajali
yeap
Sani
yesssss
bilal
hello guys
Ibitayo
when you will ask the question
Ana
anybody can ask here
bichu
is free energy possible with magnets?
joel
no
Mr.
you could construct an aparatus that might have a slightly higher 'energy profit' than energy used, but you would havw to maintain the machine, and most likely keep it in a vacuum, for no air resistance, and cool it, so chances are quite slim.
Mr.
calculate the force, p, required to just make a 6kg object move along the horizontal surface where the coefficient of friction is 0.25
Gbolahan
Yes ask
Albert
if a man travel 7km 30degree east of North then 10km east find the resultant displacement
Ajali Reply
11km
Dohn
disagree. Displacement is the hypotenuse length of the final position to the starting position. Find x,y components of each leg of journey to determine final position, then use final components to calculate the displacement.
Daniel
1.The giant star Betelgeuse emits radiant energy at a rate of 10exponent4 times greater than our sun, where as it surface temperature is only half (2900k) that of our sun. Estimate the radius of Betelgeuse assuming e=1, the sun's radius is s=7*10exponent8metres
James Reply
2. A ceramic teapot (e=0.20) and a shiny one (e=0.10), each hold 0.25 l of at 95degrees. A. Estimate the temperature rate of heat loss from each B. Estimate the temperature drop after 30mins for each. Consider only radiation and assume the surrounding at 20degrees
James
Practice Key Terms 3

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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