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  • Compare simple harmonic motion with uniform circular motion.

The figure shows a clock-wise rotating empty merry go round with iron bars holding the decorated wooden horse statues, four in each column.
The horses on this merry-go-round exhibit uniform circular motion. (credit: Wonderlane, Flickr)

There is an easy way to produce simple harmonic motion by using uniform circular motion. [link] shows one way of using this method. A ball is attached to a uniformly rotating vertical turntable, and its shadow is projected on the floor as shown. The shadow undergoes simple harmonic motion. Hooke’s law usually describes uniform circular motions ( ω size 12{ω} {} constant) rather than systems that have large visible displacements. So observing the projection of uniform circular motion, as in [link] , is often easier than observing a precise large-scale simple harmonic oscillator. If studied in sufficient depth, simple harmonic motion produced in this manner can give considerable insight into many aspects of oscillations and waves and is very useful mathematically. In our brief treatment, we shall indicate some of the major features of this relationship and how they might be useful.

The given figure shows a vertical turntable with four floor projecting light bulbs at the top. A smaller sized rectangular bar is attached to this turntable at the bottom half, with a circular knob attached to it. A red colored small ball is rolling along the boundary of this knob in angular direction, and the lights falling through this ball are ball making shadows just under the knob on the floor. The middle shadow is the brightest and starts fading as we look through to the cornered shadow.
The shadow of a ball rotating at constant angular velocity ω size 12{ω} {} on a turntable goes back and forth in precise simple harmonic motion.

[link] shows the basic relationship between uniform circular motion and simple harmonic motion. The point P travels around the circle at constant angular velocity ω size 12{ω} {} . The point P is analogous to an object on the merry-go-round. The projection of the position of P onto a fixed axis undergoes simple harmonic motion and is analogous to the shadow of the object. At the time shown in the figure, the projection has position x size 12{x} {} and moves to the left with velocity v size 12{v} {} . The velocity of the point P around the circle equals v ¯ max size 12{ {overline {v}} rSub { size 8{"max"} } } {} .The projection of v ¯ max size 12{ {overline {v}} rSub { size 8{"max"} } } {} on the x size 12{x} {} -axis is the velocity v size 12{v} {} of the simple harmonic motion along the x size 12{x} {} -axis.

The figure shows a point P moving through the circumference of a circle in an angular way with angular velocity omega. The diameter is projected along the x axis, with point P making an angle theta at the centre of the circle. A point along the diameter shows the projection of the point P with a dotted perpendicular line from P to this point, the projection of the point is given as v along the circle and its velocity v subscript max, over the top of the projection arrow in an upward left direction.
A point P moving on a circular path with a constant angular velocity ω size 12{ω} {} is undergoing uniform circular motion. Its projection on the x-axis undergoes simple harmonic motion. Also shown is the velocity of this point around the circle, v ¯ max size 12{ {overline {v}} rSub { size 8{"max"} } } {} , and its projection, which is v size 12{v} {} . Note that these velocities form a similar triangle to the displacement triangle.

To see that the projection undergoes simple harmonic motion, note that its position x size 12{x} {} is given by

x = X cos θ , size 12{x=X"cos"θ","} {}

where θ = ω t size 12{θ=ω`t} {} , ω size 12{ω} {} is the constant angular velocity, and X size 12{X} {} is the radius of the circular path. Thus,

x = X cos ω t . size 12{x=X"cos"ω`t} {}

The angular velocity ω size 12{ω} {} is in radians per unit time; in this case size 12{2π} {} radians is the time for one revolution T size 12{T} {} . That is, ω = / T size 12{ω=2π/T} {} . Substituting this expression for ω size 12{ω} {} , we see that the position x size 12{x} {} is given by:

x ( t ) = cos t T . size 12{x \( t \) ="cos" left ( { {2π`t} over {T} } right )} {}

This expression is the same one we had for the position of a simple harmonic oscillator in Simple Harmonic Motion: A Special Periodic Motion . If we make a graph of position versus time as in [link] , we see again the wavelike character (typical of simple harmonic motion) of the projection of uniform circular motion onto the x size 12{x} {} -axis.

The given figure shows a vertical turntable with four floor projecting light bulbs at the top. A smaller sized rectangular bar is attached to this turntable at the bottom half, with a circular knob attached to it. A red colored small ball is rolling along the boundary of this knob in angular direction. The turnaround table is put upon a roller paper sheet, on which the simple harmonic motion is measured, which is shown here in oscillating waves on the paper sheet in front of the table. A graph of amplitude versus time is also given alongside the figure.
The position of the projection of uniform circular motion performs simple harmonic motion, as this wavelike graph of x size 12{x} {} versus t size 12{x} {} indicates.

Now let us use [link] to do some further analysis of uniform circular motion as it relates to simple harmonic motion. The triangle formed by the velocities in the figure and the triangle formed by the displacements ( X , x , size 12{X,x,} {} and X 2 x 2 size 12{ sqrt {X rSup { size 8{2} } - x rSup { size 8{2} } } } {} ) are similar right triangles. Taking ratios of similar sides, we see that

v v max = X 2 x 2 X = 1 x 2 X 2 . size 12{ { {v} over {v rSub { size 8{"max"} } } } = { { sqrt {X rSup { size 8{2} } - x rSup { size 8{2} } } } over {X} } = sqrt {1 - { {x rSup { size 8{2} } } over {X rSup { size 8{2} } } } } } {}

We can solve this equation for the speed v size 12{v} {} or

v = v max 1 x 2 X 2 . size 12{v=v rSub { size 8{"max"} } sqrt {1 - { {x rSup { size 8{2} } } over {X rSup { size 8{2} } } } } } {}

This expression for the speed of a simple harmonic oscillator is exactly the same as the equation obtained from conservation of energy considerations in Energy and the Simple Harmonic Oscillator .You can begin to see that it is possible to get all of the characteristics of simple harmonic motion from an analysis of the projection of uniform circular motion.

Finally, let us consider the period T size 12{T} {} of the motion of the projection. This period is the time it takes the point P to complete one revolution. That time is the circumference of the circle X size 12{2πX} {} divided by the velocity around the circle, v max size 12{v rSub { size 8{"max"} } } {} . Thus, the period T size 12{T} {} is

T = 2πX v max . size 12{T= { {2πX} over {v rSub { size 8{"max"} } } } } {}

We know from conservation of energy considerations that

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

Solving this equation for X / v max size 12{X/v rSub { size 8{"max"} } } {} gives

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

Substituting this expression into the equation for T size 12{T} {} yields

T = m k . size 12{T=2π sqrt { { {m} over {k} } } "."} {}

Thus, the period of the motion is the same as for a simple harmonic oscillator. We have determined the period for any simple harmonic oscillator using the relationship between uniform circular motion and simple harmonic motion.

Some modules occasionally refer to the connection between uniform circular motion and simple harmonic motion. Moreover, if you carry your study of physics and its applications to greater depths, you will find this relationship useful. It can, for example, help to analyze how waves add when they are superimposed.

Identify an object that undergoes uniform circular motion. Describe how you could trace the simple harmonic motion of this object as a wave.

A record player undergoes uniform circular motion. You could attach dowel rod to one point on the outside edge of the turntable and attach a pen to the other end of the dowel. As the record player turns, the pen will move. You can drag a long piece of paper under the pen, capturing its motion as a wave.

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Section summary

A projection of uniform circular motion undergoes simple harmonic oscillation.

Problems&Exercises

(a)What is the maximum velocity of an 85.0-kg person bouncing on a bathroom scale having a force constant of 1 . 50 × 10 6 N/m size 12{1 "." "50" times "10" rSup { size 8{5} } "N/m"} {} , if the amplitude of the bounce is 0.200 cm? (b)What is the maximum energy stored in the spring?

a). 0.266 m/s

b). 3.00 J

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A novelty clock has a 0.0100-kg mass object bouncing on a spring that has a force constant of 1.25 N/m. What is the maximum velocity of the object if the object bounces 3.00 cm above and below its equilibrium position? (b) How many joules of kinetic energy does the object have at its maximum velocity?

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At what positions is the speed of a simple harmonic oscillator half its maximum? That is, what values of x / X size 12{x/X} {} give v = ± v max / 2 size 12{v= +- v rSub { size 8{"max"} } /2} {} , where X size 12{X} {} is the amplitude of the motion?

± 3 2 size 12{ +- { { sqrt {3} } over {2} } } {}

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A ladybug sits 12.0 cm from the center of a Beatles music album spinning at 33.33 rpm. What is the maximum velocity of its shadow on the wall behind the turntable, if illuminated parallel to the record by the parallel rays of the setting Sun?

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Questions & Answers

2 forces whose resultant is 100N, are at right angle to each other .if one of them makes an angle of 30 degree with the resultant determine it's magnitude
Victor Reply
50 N... (50 *1.732)N
Sahim
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Sahim
Is earth is an inertial frame?
Sahim Reply
The abacus (plural abaci or abacuses), also called a counting frame, is a calculating tool that was in use in Europe, China and Russia, centuries before the adoption of the written Hindu–Arabic numeral system
Sahim
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Irungu
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Sahim
Hey.. I've a question.
Sahim Reply
Is earth inertia frame?
Sahim
only the center
Shii
What is an abucus?
Irungu
what would be the correct interrogation "what is time?" or "how much has your watch ticked?"
prakash Reply
a load of 20N on a wire of cross sectional area 8×10^-7m produces an extension of 10.4m. calculate the young modules of the material of the wire is of length 5m
Ebenezer Reply
Young's modulus = stress/strain strain = extension/length (x/l) stress = force/area (F/A) stress/strain is F l/A x
El
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Ebenezer
two bodies x and y start from rest and move with uniform acceleration of a and 4a respectively. if the bodies cover the same distance in terms of tx and ty what is the ratio of tx to ty
Oluwatola Reply
what is cesium atoms?
prakash Reply
The atoms which form the element Cesium are known as Cesium atoms.
Naman
A material that combines with and removes trace gases from vacuum tubes.
Shankar
what is difference between entropy and heat capacity
Varun
Heat capacity can be defined as the amount of thermal energy required to warm the sample by 1°C. entropy is the disorder of the system. heat capacity is high when the disorder is high.
Chathu
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Vinodhini Reply
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Vinodhini
try to imagine everything you study in 3d
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What is realm
Vinodhini Reply
The quantum realm, also called the quantum scale, is a term of art inphysics referring to scales where quantum mechanical effects become important when studied as an isolated system. Typically, this means distances of 100 nanometers (10−9meters) or less or at very low temperature.
revolutionary
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by reading it
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understanding difficult
Vinodhini
vinodhini mam, physics is used in our day to day life in all events..... everything happening around us can be explained in the base of physics..... saying simple stories happening in our daily life and relating it to physics and questioning students about how or why its happening like that can make
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How to understand easily
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even when you see this message in your phone...it works accord to a physics principle. you touch screen works based on physics, your internet works based on physics, etc....... check out google and search for it
revolutionary
what is mean by Newtonian principle of Relativity? definition and explanation with example
revolutionary Reply
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Lawal
c=1/c1+c2/1+c3 what is the answer
Akinbulejo Reply
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smith
This may seem like a really stupid question, but is mechanical energy the same as potential energy? If not, what is the difference?
Nikki Reply
what is c=1\c1,c=2\c2,c=3\c3
Akinbulejo
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Iram
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Nikki
Can someone possibly walk me through this problem? " A worker drives a 0.500 kg spike into a rail tie with a 2.50 kg sledgehammer. The hammer hits the spike with a speed of 65.0 m/s. If one-third Of the hammer's kinetic energy is converted to the internal energy of rhe hammer and spike.
Nikki
how much does the total internal energy increase
Nikki
you know the mass and the velocity of the hammer. therefore using the equation (mv^2)/2 you can find the kinetic energy. then take one third of this value and that will be your change in internal energy. here, the important thing is that spike is stationary so it does not contribute to initial Energ
Chathu
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Nikki
what is the formula for finding the to total capacitance in series arrangement
Austin Reply
Don't know
Ugwu
C = 1/C1+1/C2+1/C3
Nangbun
c=1/c1+1/c2+1/c3
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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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