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Block – spring system presents an approximation of SHM. It is assumed that spring force follows Hooke’s law and there is no dissipating forces like friction or air drag. The arrangement consisting of spring with block attached at one end has many interesting variations. We analyze all such arrangements following certain basic steps to arrive at formulations for periodic attributes like time period and frequency etc.

The first step is to drive an equation between force and displacement or between acceleration and displacement. We, then, use standard expression to determine time period and frequency. Broadly, we shall be working to analyze following variations consisting of a block and spring(s) :

  • Horizontal block – spring system
  • Vertical block – spring system
  • Block connected to springs in series
  • Block in between two springs
  • Block connected to springs in parallel

Horizontal block-spring system

The Hooke’s law governing an ideal spring relates spring force with displacement as :

F = - k x

Horizontal block-spring system

The spring is stretched a bit and then let go to oscillate.

Combining with Newton’s second law,

F = m a = - k x

a = - k m x

Now, comparing with SHM relation “ a = - ω 2 x ”, we have :

ω = k m T = 2 π m k ν = 1 2 π k m

In these expressions “m” and “k” represent inertia and spring factor respectively.

Vertical block-spring system

Vertical block-spring system differs to horizontal arrangement in the application of gravitational force. In horizontal orientation, gravitational force is perpendicular to motion and as such it is not considered for the analysis. In vertical orientation, however, the spring is in extended position due to the weight of the block before the block is set in SHM. It is in equilibrium in the extended position under the action of gravitational and spring force.

Clearly, the center of oscillation is the position of equilibrium. The block oscillates about the extended position – not about the position of neutral spring length as in the case of horizontal arrangement. Let us consider that the spring is extended by a vertical length “ y 0 ” from neutral position when it is in equilibrium position. For a further extension “y” in spring, the spring force on the block is equal to the product of spring constant and total displacement from the neutral position,

F = - k y 0 + y

Vertical block-spring system

The spring is stretched a bit from the equilibrium position and then let go to oscillate.

Note that we have considered downward displacement as positive. The spring force acting upward is opposite to displacement and hence negative. In this case, however, the net restoring force on the block is equal to the resultant of spring force acting upwards and gravity acting downward. Considering downward direction as positive,

F net = F + m g = - k y 0 + y + m g

But, for equilibrium position, we have the following relation,

m g = - k y 0

Substituting this relation in the expression of net restoring force, we have :

F net = - k y

The important point to realize here is that net restoring force is independent of gravity. It is equal to differential spring force for the additional extension – not the spring force for the total extension from the neutral position. Now, according to Newton’s second law of motion, the net restoring force is equal to the product of mass of the block and acceleration,

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
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.
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Damian Reply
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
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I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
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I'm interested in nanotube
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Source:  OpenStax, Oscillation and wave motion. OpenStax CNX. Apr 19, 2008 Download for free at http://cnx.org/content/col10493/1.12
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