# 16.3 Simple harmonic motion: a special periodic motion  (Page 3/7)

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If a time-exposure photograph of the bouncing car were taken as it drove by, the headlight would make a wavelike streak, as shown in [link] . Similarly, [link] shows an object bouncing on a spring as it leaves a wavelike "trace of its position on a moving strip of paper. Both waves are sine functions. All simple harmonic motion is intimately related to sine and cosine waves.

The displacement as a function of time t in any simple harmonic motion—that is, one in which the net restoring force can be described by Hooke’s law, is given by

$x\left(t\right)=X\phantom{\rule{0.25em}{0ex}}\text{cos}\frac{2\mathrm{\pi t}}{T},$

where $X$ is amplitude. At $t=0$ , the initial position is ${x}_{0}=X$ , and the displacement oscillates back and forth with a period $T$ . (When $t=T$ , we get $x=X$ again because $\text{cos}\phantom{\rule{0.25em}{0ex}}2\pi =1$ .). Furthermore, from this expression for $x$ , the velocity $v$ as a function of time is given by:

$v\left(t\right)=-{v}_{\text{max}}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\left(\frac{2\pi t}{T}\right),$

where ${v}_{\text{max}}=2\pi X/T=X\sqrt{k/m}$ . The object has zero velocity at maximum displacement—for example, $v=0$ when $t=0$ , and at that time $x=X$ . The minus sign in the first equation for $v\left(t\right)$ 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 $x\left(t\right),\phantom{\rule{0.25em}{0ex}}v\left(t\right),\phantom{\rule{0.25em}{0ex}}t,$ and $a\left(t\right)$ , the quantities needed for kinematics and a description of simple harmonic motion.] According to Newton’s second law, the acceleration is $a=F/m=\text{kx}/m$ . So, $a\left(t\right)$ is also a cosine function:

$a\left(t\right)=-\frac{\text{kX}}{m}\text{cos}\frac{2\pi t}{T}.$

Hence, $a\left(t\right)$ is directly proportional to and in the opposite direction to $x\left(t\right)$ .

[link] shows the simple harmonic motion of an object on a spring and presents graphs of $x\left(t\right),v\left(t\right),$ and $a\left(t\right)$ versus time.

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.

#### Questions & Answers

how can I read physics...am finding it difficult to understand...pls help
rerry Reply
try to read several books on phy don't just rely one. some authors explain better than other.
Ju
And don't forget to check out YouTube videos on the subject. Videos offer a different visual way to learn easier.
Ju
hope that helps
Ju
I have a exam on 12 february
David Reply
what is velocity
Jiti
the speed of something in a given direction.
Ju
what is a magnitude in physics
Jiti Reply
Propose a force standard different from the example of a stretched spring discussed in the text. Your standard must be capable of producing the same force repeatedly.
Giovani Reply
What is meant by dielectric charge?
It's Reply
what happens to the size of charge if the dielectric is changed?
Brhanu Reply
omega= omega not +alpha t derivation
Provakar Reply
u have to derivate it respected to time ...and as w is the angular velocity uu will relace it with "thita × time""
Abrar
do to be peaceful with any body
Brhanu Reply
the angle subtended at the center of sphere of radius r in steradian is equal to 4 pi how?
Saeed Reply
if for diatonic gas Cv =5R/2 then gamma is equal to 7/5 how?
Saeed
define variable velocity
Ali Reply
displacement in easy way.
Mubashir Reply
binding energy per nucleon
Poonam Reply
why God created humanity
Manuel Reply
Because HE needs someone to dominate the earth (Gen. 1:26)
Olorunfemi
why god made humenity
Ali
and he to multiply
Owofemi
stuff happens
Ju
God plays dice
Ju
Is the object in a conductor or an insulator? Justify your answer. whats the answer to this question? pls need help figure is given above
Jun Reply
ok we can say body is electrically neutral ...conductor this quality is given to most metalls who have free electron in orbital d ...but human doesn't have ...so we re made from insulator or dielectric material ... furthermore, the menirals in our body like k, Fe , cu , zn
Abrar
when we face electric shock these elements work as a conductor that's why we got this shock
Abrar
how do i calculate the pressure on the base of a deposit if the deposit is moving with a linear aceleration
ximena Reply

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