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  • Determine the maximum speed of an oscillating system.

To study the energy of a simple harmonic oscillator, we first consider all the forms of energy it can have We know from Hooke’s Law: Stress and Strain Revisited that the energy stored in the deformation of a simple harmonic oscillator is a form of potential energy given by:

PE el = 1 2 kx 2 . size 12{"PE" size 8{"el"}= { {1} over {2} } ital "kx" rSup { size 8{2} } } {}

Because a simple harmonic oscillator has no dissipative forces, the other important form of energy is kinetic energy KE size 12{ ital "KE"} {} . Conservation of energy for these two forms is:

KE + PE el = constant size 12{ ital "KE"+ ital "PE" rSub { size 8{e1} } ="constant"} {}

or

1 2 mv 2 + 1 2 kx 2 = constant. size 12{ { {1} over {2} } ital "mv" rSup { size 8{2} } + { {1} over {2} } ital "kx" rSup { size 8{2} } ="constant"} {}

This statement of conservation of energy is valid for all simple harmonic oscillators, including ones where the gravitational force plays a role

Namely, for a simple pendulum we replace the velocity with v = size 12{v=Lω} {} , the spring constant with k = mg / L size 12{k= ital "mg"/L} {} , and the displacement term with x = size 12{x=Lθ} {} . Thus

1 2 mL 2 ω 2 + 1 2 mgL θ 2 = constant. size 12{ { {1} over {2} } ital "mL" rSup { size 8{2} } ω rSup { size 8{2} } + { {1} over {2} } ital "mgL"θ rSup { size 8{2} } ="constant"} {}

In the case of undamped simple harmonic motion, the energy oscillates back and forth between kinetic and potential, going completely from one to the other as the system oscillates. So for the simple example of an object on a frictionless surface attached to a spring, as shown again in [link] , the motion starts with all of the energy stored in the spring. As the object starts to move, the elastic potential energy is converted to kinetic energy, becoming entirely kinetic energy at the equilibrium position. It is then converted back into elastic potential energy by the spring, the velocity becomes zero when the kinetic energy is completely converted, and so on. This concept provides extra insight here and in later applications of simple harmonic motion, such as alternating current circuits.

Figure a shows a spring on a frictionless surface attached to a bar or wall from the left side, and on the right side of it there’s an object attached to it with mass m, its amplitude is given by X, and x equal to zero at the equilibrium level. Force F is applied to it from the right side, shown with left direction pointed red arrow and velocity v is equal to zero. A direction point showing the north and west direction is also given alongside this figure as well as with other four figures. The energy given here for the object is given according to the velocity. In figure b, after the force has been applied, the object moves to the left compressing the spring a bit, and the displaced area of the object from its initial point is shown in sketched dots. F is equal to zero and the V is max in negative direction. The energy given here for the object is given according to the velocity. In figure c, the spring has been compressed to the maximum level, and the amplitude is negative x. Now the direction of force changes to the rightward direction, shown with right direction pointed red arrow and the velocity v zero. The energy given here for the object is given according to the velocity.                In figure d, the spring is shown released from the compressed level and the object has moved toward the right side up to the equilibrium level. F is zero, and the velocity v is maximum. The energy given here for the object is given according to the velocity.               In figure e, the spring has been stretched loose to the maximum level and the object has moved to the far right. Now again the velocity here is equal to zero and the direction of force again is to the left hand side, shown here as F is equal to zero. The energy given here for the object is given according to the velocity.
The transformation of energy in simple harmonic motion is illustrated for an object attached to a spring on a frictionless surface.

The conservation of energy principle can be used to derive an expression for velocity v size 12{v} {} . If we start our simple harmonic motion with zero velocity and maximum displacement ( x = X size 12{x=X} {} ), then the total energy is

1 2 kX 2 . size 12{ { {1} over {2} } ital "kX" rSup { size 8{2} } } {}

This total energy is constant and is shifted back and forth between kinetic energy and potential energy, at most times being shared by each. The conservation of energy for this system in equation form is thus:

1 2 mv 2 + 1 2 kx 2 = 1 2 kX 2 . size 12{ { {1} over {2} } ital "mv" rSup { size 8{2} } + { {1} over {2} } ital "kx" rSup { size 8{2} } = { {1} over {2} } ital "kX" rSup { size 8{2} } } {}

Solving this equation for v size 12{v} {} yields:

v = ± k m X 2 x 2 . size 12{v= +- sqrt { { {k} over {m} } left (X rSup { size 8{2} } - x rSup { size 8{2} } right )} } {}

Manipulating this expression algebraically gives:

v = ± k m X 1 x 2 X 2 size 12{v= +- sqrt { { {k} over {m} } } X sqrt {1 - { {x rSup { size 8{2} } } over {X rSup { size 8{2} } } } } } {}

and so

v = ± v max 1 x 2 X 2 , size 12{v= +- v size 8{"max" sqrt {1 - { {x rSup { size 8{2} } } over {X rSup { size 8{2} } } } } }} {}

where

v max = k m X . size 12{v size 8{"max"}= sqrt { { {k} over {m} } } X} {}

From this expression, we see that the velocity is a maximum ( v max ) at x = 0 size 12{x=0} {} , as stated earlier in v t = v max sin t T . Notice that the maximum velocity depends on three factors. Maximum velocity is directly proportional to amplitude. As you might guess, the greater the maximum displacement the greater the maximum velocity. Maximum velocity is also greater for stiffer systems, because they exert greater force for the same displacement. This observation is seen in the expression for v max ; it is proportional to the square root of the force constant k . Finally, the maximum velocity is smaller for objects that have larger masses, because the maximum velocity is inversely proportional to the square root of m . For a given force, objects that have large masses accelerate more slowly.

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, General physics ii phy2202ca. OpenStax CNX. Jul 05, 2013 Download for free at http://legacy.cnx.org/content/col11538/1.2
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