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  • Compare and discuss underdamped and overdamped oscillating systems.
  • Explain critically damped system.
In the figure shown, a small child is seated in a spring swing, tied with a belt at his waist. In the back is his father, who is pushing the swing in the to and fro motion.
In order to counteract dampening forces, this dad needs to keep pushing the swing. (credit: Erik A. Johnson, Flickr)

A guitar string stops oscillating a few seconds after being plucked. To keep a child happy on a swing, you must keep pushing. Although we can often make friction and other non-conservative forces negligibly small, completely undamped motion is rare. In fact, we may even want to damp oscillations, such as with car shock absorbers.

For a system that has a small amount of damping, the period and frequency are nearly the same as for simple harmonic motion, but the amplitude gradually decreases as shown in [link] . This occurs because the non-conservative damping force removes energy from the system, usually in the form of thermal energy. In general, energy removal by non-conservative forces is described as

W nc = Δ ( KE + PE ) , size 12{W rSub { size 8{ ital "nc"} } =Δ \( ital "KE"+ ital "PE" \) ","} {}

where W nc size 12{W rSub { size 8{ ital "nc"} } } {} is work done by a non-conservative force (here the damping force). For a damped harmonic oscillator, W nc size 12{W rSub { size 8{ ital "nc"} } } {} is negative because it removes mechanical energy (KE + PE) from the system.

The figure shows a graph of displacement, along y axis, versus time for a harmonic oscillator, running along the x axis, equally in positive as well as negative quadrants. The amplitude, X, of the oscillations is shown by a green wave which is decreasing gradually as we move far from the y axis, but the time, T, shown here as the distance between adjacent crests remains the same throughout.
In this graph of displacement versus time for a harmonic oscillator with a small amount of damping, the amplitude slowly decreases, but the period and frequency are nearly the same as if the system were completely undamped.

If you gradually increase the amount of damping in a system, the period and frequency begin to be affected, because damping opposes and hence slows the back and forth motion. (The net force is smaller in both directions.) If there is very large damping, the system does not even oscillate—it slowly moves toward equilibrium. [link] shows the displacement of a harmonic oscillator for different amounts of damping. When we want to damp out oscillations, such as in the suspension of a car, we may want the system to return to equilibrium as quickly as possible Critical damping is defined as the condition in which the damping of an oscillator results in it returning as quickly as possible to its equilibrium position The critically damped system may overshoot the equilibrium position, but if it does, it will do so only once. Critical damping is represented by Curve A in [link] . With less-than critical damping, the system will return to equilibrium faster but will overshoot and cross over one or more times. Such a system is underdamped ; its displacement is represented by the curve in [link] . Curve B in [link] represents an overdamped system. As with critical damping, it too may overshoot the equilibrium position, but will reach equilibrium over a longer period of time.

The given graph is of displacement, along y-axis, versus time along x axis. Symbol for displacement is given as X subscript zero and for time is given as t. Two curves start at a point along the y axis, where X subscript zero is greater than zero. Curve A is curved downward and meets x axis at a point. Curve B is curved upward and is over curve A and meets x axis at a point which is toward the far right of the meeting point of the curve A and x axis.
Displacement versus time for a critically damped harmonic oscillator (A) and an overdamped harmonic oscillator (B). The critically damped oscillator returns to equilibrium at X = 0 size 12{X=0} {} in the smallest time possible without overshooting.

Critical damping is often desired, because such a system returns to equilibrium rapidly and remains at equilibrium as well. In addition, a constant force applied to a critically damped system moves the system to a new equilibrium position in the shortest time possible without overshooting or oscillating about the new position. For example, when you stand on bathroom scales that have a needle gauge, the needle moves to its equilibrium position without oscillating. It would be quite inconvenient if the needle oscillated about the new equilibrium position for a long time before settling. Damping forces can vary greatly in character. Friction, for example, is sometimes independent of velocity (as assumed in most places in this text). But many damping forces depend on velocity—sometimes in complex ways, sometimes simply being proportional to velocity.

Questions & Answers

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Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
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Anassong Reply
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s.
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Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
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CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
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s.
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for screen printed electrodes ?
SUYASH
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s. Reply
of graphene you mean?
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s.
Graphene has a hexagonal structure
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Source:  OpenStax, Yupparaj english program physics corresponding to thai physics book #3. OpenStax CNX. May 19, 2014 Download for free at http://legacy.cnx.org/content/col11657/1.1
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