# 16.8 Forced oscillations and resonance  (Page 2/5)

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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.

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.

## 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?

## 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

If a car has a suspension system with a force constant of $5\text{.}\text{00}×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{N/m}$ , how much energy must the car’s shocks remove to dampen an oscillation starting with a maximum displacement of 0.0750 m?

(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.

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 ${\mu }_{\text{s}}=0\text{.}\text{100}$ . (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 ${\mu }_{\text{k}}=0\text{.}\text{0850}$ , what total distance does it travel before stopping? Assume it starts at the maximum amplitude.

Engineering Application: A suspension bridge oscillates with an effective force constant of $1\text{.}\text{00}×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{N/m}$ . (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 $1\text{.}\text{00}×{\text{10}}^{4}\phantom{\rule{0.25em}{0ex}}\text{J}$ 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) $5\text{.}\text{00}×{\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\text{J}$

(b) $\text{1.20}\text{×}{\text{10}}^{3}$ s

what is power
Power is the rate at which work is done with respect to time.
Prinze
power is ratio work to time
babar
power of a body is its energy possessed in unit time
babar
how much work can be done is within smallest time frame is the power p=w and p=1/T !
Mudang
Work done by a body in per unit time is Power... Simple
Aayush
work done per unit time is power which can be expressed in wattt or horsepower unit simply
Saeed
Yes ✔️ Power (P) = work done by time taken... ie. P= W/T
Aayush
Power can be explained physically as speed of energy change
power is the dot product of force and velocity
sana
A bicycle has a diameter of 63cm.calculate how many times the wheel turns round intravelling 19.8km.use the value 31/7 for tether
What is torque?
the cross product of force and position vector
the angular analouge of force
the product of one force and the perpendicular distance between forces
Lekunga
t=bil
Lekunga
where b=flux density
Lekunga
hey guys I'm new here
Djhope
rotational effect of force
babar
the angular analogue of F=ma would be Tao=I•alpha where I is the moment of inertia and alpha is angular acceleration
Julia
what is physical quality?
it can be measure directly or indirectly.
NARENDRA
a quantity, by definition has a number associated with it, and can be measured as you say. a quality isn't necessarily a number representing something. qualities do not need to be quantitative.
Julia
physical qualities are characteristic of specifics of a particular object, for examples a 500ml 1st container contains 500ml of water and 10.5ml of NACL and 2nd container contain 500ml of water and 0.5ml NACL the Difference between Two of them is the Quality of 2
Mudang
if we buy some thing (cloth) we always not satisfied..... unless some one compare with other things that is quality,we buy that thing
babar
we buy that thing after our satisfaction.
babar
what is a wave
A wave is a disturbance that carry energy from one place to another.
what is the importance of electric and magnetic field in mass spectrograph ?
you can determine the specific charge of a particle from the radius of gyration (Lamar radius) in the magnetic field.
Sean
furthermore the electric field accelerates the charges, so you can determine the kinetic energy etc
Sean
what is motion
a motion is the change in position of an object copmaring to a fixed point
ahmed
comparing*
ahmed
moving or move from one place to another place
Boomi
motion is the movement of an object from one place to another
David
Motion is the continues movement of an object from one fixed point to another with respect to time.
LAHBAN
what is acceleration: this is when the rate of an object changes in respect to it surrounding with time.
Turay
acceleration is change in velocity over time
mohammed
motion doesn't necessarily need to include a change in acceleration: motion is a change in position.
Julia
change in position
Azeem
Motion is the change of position from one point in a space to other point in the same space.. sorry for my english I'm francophone...
André-Franck
motion isn't really a scientific word so much. it just means literally how an object moves. İf you want to describe the motion of an object, to give a full picture you need to tell me what an object is at any moment of time I ask. From there, I can deduce anything I want: velocity, acceleration
Sean
Please keep in mind that it's not allowed to promote any social groups (whatsapp, facebook, etc...), exchange phone numbers, email addresses or ask for personal information on our platform.
ok
Chelsea
ok
babar
can u explain the term displacement in another form
displacement is the change in position of an object
Owusu
this is the a average distance covered
Lekunga
change in position with respect to reference point
babar
resultant of position vectors or difference of position vectors of a moving object
babar
Distance with a directional component.
Emmanuel
displacement is the difference between your final and initial positions independent of the path taken. distance is how far you actually moved.
Julia
it is an arrow on a map that can't be rotated or stretched, but it can be moved around. İf I place the tail of it on where you started a journey, the pointy end must lie EXACTLY on where you finished your journey.
Sean
what is gravity
no one really knows
Sean
how much do you need to know? what are you studying about it?
Sean
humans like to always point to a cause of something. İn fact, we have a phrase in English: "Everything happens for a reason". So, we have noticed ever since we have been alive that any object that had ever existed seems to be pulled by some "invisible hand" towards the earth.
Sean
And also the moon and earth "pull on eachother"...
Sean
Newton came up with this breakthrough that the type of pull between the moon and the earth is the same type of pull between the famous apple and the earth, and that this phenomenon is EVERYWHERE- universal!
Sean
and amazingly, we can actually predict exactly how this "hand" will pull on things.
Sean
Many many years later, Einstein came along and thought of gravity not like some invisible hand but more like something that is part of space and time itself... General relativity!
Sean
is the acceleration ever in the same direction as a component of velocity
what do you mean?
Sean
explain what "component" means to me
Sean
A 12V battery has a 1ohm resistor in series with a parallel system of 3ohms and 6ohms. How much current is going through the 3ohm resistor?
Total circuit current, I=12/3=4 amperes this means that 8v will be across the parallel resistors. Current through the 3 ohms is 8/3 amperes
Barnabas
How do you know there is 8v across the parallel resistors?
Ingrid
good question
Lekunga
Lekunga
3//6 combined is 2 ohms. Total cct R is 3 ohms, I=V/R=4amps. V across 3//6 is therefore = I x R = 8 volts.
Peter
thanks
Lekunga
a particle f suspended by two strings passing over smooth pulley isattached to two other particles q and r .. calculate the masses of p and q if mass of r is 2 kg
there isn't enough information to get a single number for p and q. also, you didn't give information about what p was doing. did you mean particle p instead of f?
Sean
this is like saying two people sit on a seesaw and the seesaw remains horizontal, what are their masses?
Sean
the idea is that assuming when you leave the system alone nothing falls, the weight of q plus weight of r must equal weight of p
Sean
what is hall effect and what are its applications?
It's under sound wave
for speed detection.i.e speed of current
Ben
speed of the current is nothing but voltage
sharan
the hall effect is used to measure a magnetic field strength
Sean
This is to do with the fact that current responds to a magnetic field. the charges start turning and end up on one side of the wire. this seperation of charge then ends up causing something you can actually measure, called a potential difference, (or equivalently) a voltage.
Sean
@sharan that is not correct. speed of charges is current. speed of current is... current, as when you say speed of current you are saying "speed of speed of charges". Voltage is how much energy is transferred to one unit of charge (the unit being Coloumb in Sİ units)
Sean