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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
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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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