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v ( t ) = v max sin t T , size 12{v \( t \) = - v rSub { size 8{"max"} } "sin" left ( { {2π`t} over {T} } right )} {}

where v max = X / T = X k / m size 12{v rSub { size 8{"max"} } =2πX/T=X sqrt {k/m} } {} . The object has zero velocity at maximum displacement—for example, v = 0 size 12{v=0} {} when t = 0 size 12{t=0} {} , and at that time x = X size 12{x=X} {} . The minus sign in the first equation for v ( t ) size 12{v \( t \) } {} gives the correct direction for the velocity. Just after the start of the motion, for instance, the velocity is negative because the system is moving back toward the equilibrium point. Finally, we can get an expression for acceleration using Newton’s second law. [Then we have x ( t ) , v ( t ) , t , size 12{x \( t \) ,v \( t \) ,t} {} and a ( t ) size 12{a \( t \) } {} , the quantities needed for kinematics and a description of simple harmonic motion.] According to Newton’s second law, the acceleration is a = F / m = kx / m size 12{a=F/m= ital "kx"/m} {} . So, a ( t ) size 12{a \( t \) } {} is also a cosine function:

a ( t ) = kX m cos t T . size 12{a \( t \) = - { { ital "kX"} over {m} } " cos " { {2π t} over {T} } } {}

Hence, a ( t ) size 12{a \( t \) } {} is directly proportional to and in the opposite direction to x ( t ) .

[link] shows the simple harmonic motion of an object on a spring and presents graphs of x ( t ) , v ( t ), size 12{x \( t \) ,v \( t \) `} {} and a ( t ) size 12{`a \( t \) } {} versus time.

In the figure at the top there are ten springboards with objects of different mass values tied to them. This makes some springs highly compressed some as loosely stretched and some at equilibrium, which are shown as red spherical shaped. Alongside the figure there is a scale given for different amplitude values as x equal to positive X, zero and negative X. the upward and downward pointing arrows are shown with a few springboards.  In the second figure there are three graphs. The first graph shows distance covered in form of a sine wave starting from a point x units on positive y-axis. The height of the wave above x-axis is marked as amplitude. The gap between two consecutive crests is marked as T. Below first graph there is another graph showing velocity in form of a sine wave starting from the origin downward. In the third graph below the second one, acceleration is shown in the form of sine wave starting from x units on the negative y-axis upward. In the last figure three position of a spring are shown. The first position shows the unstretched length of a spring pendulum. A hand is holding the bob of the pendulum. In the second position the equilibrium position of the spring and bob is shown. This position is lower the first one. In the third case the up and down oscillations of the spring pendulum are shown. The bob is moving x units in upward and downward directions alternatively.
Graphs of x ( t ) , v ( t ) , size 12{x \( t \) ,v \( t \) `} {} and a ( t ) size 12{`a \( t \) } {} versus t size 12{t} {} for the motion of an object on a spring. The net force on the object can be described by Hooke’s law, and so the object undergoes simple harmonic motion. Note that the initial position has the vertical displacement at its maximum value X size 12{X} {} ; v size 12{v} {} is initially zero and then negative as the object moves down; and the initial acceleration is negative, back toward the equilibrium position and becomes zero at that point.

The most important point here is that these equations are mathematically straightforward and are valid for all simple harmonic motion. They are very useful in visualizing waves associated with simple harmonic motion, including visualizing how waves add with one another.

Suppose you pluck a banjo string. You hear a single note that starts out loud and slowly quiets over time. Describe what happens to the sound waves in terms of period, frequency and amplitude as the sound decreases in volume.

Frequency and period remain essentially unchanged. Only amplitude decreases as volume decreases.

A babysitter is pushing a child on a swing. At the point where the swing reaches x size 12{x} {} , where would the corresponding point on a wave of this motion be located?

x size 12{x} {} is the maximum deformation, which corresponds to the amplitude of the wave. The point on the wave would either be at the very top or the very bottom of the curve.

Phet explorations: masses and springs

A realistic mass and spring laboratory. Hang masses from springs and adjust the spring stiffness and damping. You can even slow time. Transport the lab to different planets. A chart shows the kinetic, potential, and thermal energy for each spring.

Masses and Springs

Section summary

  • Simple harmonic motion is oscillatory motion for a system that can be described only by Hooke’s law. Such a system is also called a simple harmonic oscillator.
  • Maximum displacement is the amplitude X size 12{X} {} . The period T size 12{T} {} and frequency f size 12{f} {} of a simple harmonic oscillator are given by

    T = m k size 12{T=2π sqrt { { {m} over {k} } } } {} and f = 1 k m , where m size 12{m} {} is the mass of the system.

  • Displacement in simple harmonic motion as a function of time is given by x ( t ) = X cos t T . size 12{x \( t \) =X"cos" { {2π`t} over {T} } } {}
  • The velocity is given by v ( t ) = v max sin t T , where v max = k / m X .
  • The acceleration is found to be a ( t ) = kX m cos t T . size 12{a \( t \) = - { { ital "kX"} over {m} } " cos " { {2π t} over {T} } } {}

Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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Damian
yes that's correct
Professor
I think
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industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
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scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
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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
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if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
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analytical skills graphene is prepared to kill any type viruses .
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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?
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what king of growth are you checking .?
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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why we need to study biomolecules, molecular biology in nanotechnology?
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biomolecules are e building blocks of every organics and inorganic materials.
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Introduction about quantum dots in nanotechnology
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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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