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

 Page 3 / 7
$v\left(t\right)=-{v}_{\text{max}}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}\left(\frac{2\pi t}{T}\right),$

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.

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

$x$ 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.

## 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 $X$ . The period $T$ and frequency $f$ of a simple harmonic oscillator are given by

$T=2\pi \sqrt{\frac{m}{k}}$ and $f=\frac{1}{2\pi }\sqrt{\frac{k}{m}}$ , where $m$ is the mass of the system.

• Displacement in simple harmonic motion as a function of time is given by $x\left(t\right)=X\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\frac{2\pi t}{T}.$
• The velocity is given by $v\left(t\right)=-{v}_{\text{max}}\text{sin}\frac{2\pi \text{t}}{T}$ , where ${v}_{\text{max}}=\sqrt{k/m}X$ .
• The acceleration is found to be $a\left(t\right)=-\frac{\mathrm{kX}}{m}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\frac{2\pi t}{T}.$

I need someone to explain how white light disperses to form the "ROYGBIV".
when it pass through a glass prism through a process called dispersion of light
Mahmud
What is an atom
An atom is the smallest indivisible particle of an element
Dera
When a toilet is flushed or a sink is drained, the water (and other material) begins to rotate about the drain on the way down. Assuming no initial rotation and a flow initially directly straight toward the drain, explain what causes the rotation and which direction it has in the northern hemisphere.
find the change in entropy of a 2.00 kg block of gold at 1063^0C when it meets to become liquid gold at 1063^0C
if you are asked to make a very sensitive thermometer which of the following fluids would you choose
precious
between mercy and gasoline
precious
it good to use mercury because mercury does not wet glass and it does not evaporate easily
Desmond
0
firdaus
SFAR Sifar SIFAT -<SIFST
firdaus
how many particles are in 2 moles of chromium
if so use the normal formula number of atom= number of particle/Avogadro's number
Aki
n= np/avogadtos constant. therefore n= 24/ 6.022×10²³
albert
24÷6.022×10²³
albert
@Albert is wrong
Aki
when you cross multiple it should give you Number of particles= mole*Avogadro's number X=2m*6.022*10^²³ X=1.20*10²⁴g
Aki
1.204×10^-22
Maame
please what is final velocity and initial velocity
don't know
Ekene
what do you want to become in future
Ekene
Nonso
I think initial velocity is the velocity that the mobile starts with at the start time (t=0s) but I don't think I heard abt final velocity
Malak
Malak where are you now I need to learn more from you
Ekene
initial velocity is the velocity an object possess at it intial position or is the starting velocity, while a final velocity is the velocity an object or body possess at it final stage or at the end of it motion
Mubarak
Bohr is kimia, of toksid, cloud, tree have cloud, tree, river but small from toksid fish or another.
Heavy, heavy kehidupan susah, kekayaan, berlambak, bergumpul. Dikenali.
Gravitional, Gravitional mean kehidup seseorang. Kehidupan bumi, kehidupan muka bumi, kehidupan dalam longitude, kehidupan dalam momentom, kehidupan dalam mongitude. Kehidupan dalam Pelajaran, mean Pelajar kolej.
firdaus
Nonconservative. Sains belajar
firdaus
hello
LFX
what are the types of kinetics
what is torque pls
An aluminum rod of length 1.8cm at 0°C is heated to produced a difference in length of 0.007cm. Calculate the temperature to which it is heated. Take the linear expansively aluminun as 2.3×10^-5 K^-1.
apply the formula... linear expansivity... everything you need are in the question already
Aki
Guys pls who understand equilibrium of forces mostly the calculation aspect
Faith
so can u explain how to solve the calculations
Faith
CcmT 0°0°M°T 4°4°0°0 4400°c Right Calculate 4400c Don't have rules with him
firdaus
please can someone explain how coulombs law is used to determine electric force
u know coulomb's law and electric force are related in a formula which is E=f/q where E is equals to electric field intensity and f is equals to the force while-q is equals to the charge
Juilet
and I guess no the formula for electric field intensity
Juilet
and I guess you know the formula for electric field intensity
Juilet
exactly what i want to say
Haruna
Field mean like don't have we give have.. Yap... That oil Airplan. Dshell n petrol..
firdaus
what is momentum
Momentum mean bacteria... OR CARE MONEY FROM STOLED.. OR CARE WE PLACE OR HOME FROM STOLED OR CARE WE BODY FROM STOLED
firdaus
Moment
firdaus
let him explain the statement well
Hello guys I'm new here
diewgatdet
hello
Godwin
Statement is resit, temu bual.
firdaus
the specimens of different materials can have same if their dimensions don't match a) resistance b) resistivity c) both d) statement is ambiguous
shoukat
b
diewgatdet
Bacteria.
firdaus
Specimen mean super human.. Power hin
firdaus
, yap like that lag
firdaus