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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.

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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.

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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

Wheatstone bridge problems
Sonali Reply
what is motion
Ali Reply
When a body changes its position with respect to its surroundings then it is said to be in motion
Jon
it is a change in position, direction, shape and arrangement of a body or an object which is initially at rest
Abolarin
Pls guys am having problem on these topics: latent heat of fusion, specific heat capacity and the sub topics under them.Pls who can help?
hamidat Reply
Thanks George,I appreciate.
hamidat
this will lead you rightly of the formula to use
Abolarin
Most especially it is the calculatory aspects that is giving me issue, but with these new strength that you guys have given me,I will put in my best to understand it again.
hamidat
you can bring up a question and let's see what we can do to it
Abolarin
I also have a problem with the calculation sincerely
Sulaimon
the distance between two suasive crests of water wave traveling of 3.6ms1 is 0.45m calculate the frequency of the wave
Idris Reply
v=f×lemda where the velocity is given and lends also given so simply u can calculate the frequency
Abdul
You are right my brother, make frequency the subject of formula and equate the values of velocity and lamda into the equation, that all.
hamidat
lExplain what happens to the energy carried by light that it is dimmed by passing it through two crossed polarizing filters.
Christoper Reply
When light is reflected at Brewster's angle from a smooth surface, it is 100% polarizedparallel to the surface. Part of the light will be refracted into the surface.
Ekram
What is specific heat capacity?
hamidat Reply
Specific heat capacity is the amount of heat required to raise the temperature of one (Kg) of a substance through one Kelvin
Paluutar
formula for measuring Joules
Rowshan Reply
I don't understand, do you mean the S.I unit of work and energy?
hamidat
what are the effects of electric current
ADAMS Reply
What limits the Magnification of an optical instrument?
Naeem Reply
Lithography is 2 micron
Venkateshwarlu
what is expression for energy possessed by water ripple
Prabesh Reply
what is hydrolic press
Mark Reply
An hydraulic press is a type of machine that is operated by different pressure of water on pistons.
hamidat
what is dimensional unite of mah
Patrock Reply
i want jamb related question on this asap🙏
sharon Reply
What is Boyles law
Pascal Reply
it can simple defined as constant temperature
Muhammad
Boyles law states that the volume of a fixed amount of a gas is inversely proportional to the pressure acting on in provided that the temperature is constant.that is V=k(1/p) or V=k/p
FADILAT
what is motion
Mua Reply
getting notifications for a dictionary word, smh
Anderson
Practice Key Terms 3

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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