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Uniform Circular Motion and Simple Harmonic 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.

Figure 16.17 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. Figure 16.18 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 ( $ω ω size 12{ω} {}$ constant) rather than systems that have large visible displacements. So observing the projection of uniform circular motion, as in Figure 16.18, 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.

Figure 16.18 The shadow of a ball rotating at constant angular velocity

ω ω size 12{ω} {}

on a turntable goes back and forth in precise simple harmonic motion.

Figure 16.19 shows the basic relationship between uniform circular motion and simple harmonic motion. The point P travels around the circle at constant angular velocity $ω ω size 12{ω} {}$ . 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 $x x size 12{x} {}$ and moves to the left with velocity $v v size 12{v} {}$ . The velocity of the point P around the circle equals ${\bar{v}}_{max} {\bar{v}}_{max} size 12{ {overline {v}} rSub { size 8{"max"} } } {}$ .The projection of ${\bar{v}}_{max} {\bar{v}}_{max} size 12{ {overline {v}} rSub { size 8{"max"} } } {}$ on the $x x size 12{x} {}$ -axis is the velocity $v v size 12{v} {}$ of the simple harmonic motion along the $x x size 12{x} {}$ -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.

Figure 16.19 A point P moving on a circular path with a constant angular velocity

ω 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,

{\bar{v}}_{max} {\bar{v}}_{max} size 12{ {overline  {v}}  rSub { size 8{"max"} } } {}

, and its projection, which is

v 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 $x x size 12{x} {}$ is given by

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

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

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

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

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

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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 Figure 16.20, we see again the wavelike character (typical of simple harmonic motion) of the projection of uniform circular motion onto the $x x size 12{x} {}$ -axis.

Figure 16.20 The position of the projection of uniform circular motion performs simple harmonic motion, as this wavelike graph of

x x size 12{x} {}

versus

t t size 12{x} {}

indicates.

Now let us use Figure 16.19 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 ( $X, x, X, size 12{X,x,} {}$ and $\sqrt{X^{2} - x^{2}} \sqrt{X^{2} - x^{2}} size 12{ sqrt {X rSup { size 8{2} } - x rSup { size 8{2} } } } {}$ ) are similar right triangles. Taking ratios of similar sides, we see that

\frac{v}{v_{max}} = \frac{\sqrt{X^{2} - x^{2}}}{X} = \sqrt{1 - \frac{x^{2}}{X^{2}}} . \frac{v}{v_{max}} = \frac{\sqrt{X^{2} - x^{2}}}{X} = \sqrt{1 - \frac{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} } } } } } {}

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

v = v_{max} \sqrt{1 - \frac{x^{2}}{X^{2}}} . v = v_{max} \sqrt{1 - \frac{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} } } } } } {}

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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 $T T size 12{T} {}$ 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 $2π X 2π X size 12{2πX} {}$ divided by the velocity around the circle, $v_{max} v_{max} size 12{v rSub { size 8{"max"} } } {}$ . Thus, the period $T T size 12{T} {}$ is

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

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We know from conservation of energy considerations that

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

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

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

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

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

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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.

Check Your Understanding

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

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Solution

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.