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

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

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 $a\left(t\right)=-\frac{\mathrm{kX}}{m}\phantom{\rule{0.25em}{0ex}}\text{cos}\phantom{\rule{0.25em}{0ex}}\frac{2\pi t}{T}.$

explain how a body becomes electrically charged based on the presence of charged particles
induction
babar
induction
DEMGUE
definitely by induction
Raymond
induction
Raymond
induction
Shah
induction
Korodhso
please why does a needle sinks in water
DEMGUE
what are the calculations of Newton's third law of motiow
what is dark matter
(in some cosmological theories) non-luminous material which is postulated to exist in space and which could take either of two forms: weakly interacting particles ( cold dark matter ) or high-energy randomly moving particles created soon after the Big Bang ( hot dark matter ).
Usman
if the mass of a trolley is 0.1kg. calculate the weight of plasticine that is needed to compensate friction. (take g=10m/s and u=0.2)
what is a galaxy
what isflow rate of volume
flow rate is the volume of fluid which passes per unit time;
Rev
flow rate or discharge represnts the flow passing in unit volume per unit time
bhat
When two charges q1 and q2 are 6 and 5 coulomb what is ratio of force
When reducing the mass of a racing bike, the greatest benefit is realized from reducing the mass of the tires and wheel rims. Why does this allow a racer to achieve greater accelerations than would an identical reduction in the mass of the bicycle’s frame?
nehemiah
why is it proportional
i don't know
y
nehemiah
what are the relationship between distance and displacement
They are interchangeable.
Shii
Distance is scalar, displacement is vector because it must involve a direction as well as a magnitude. distance is the measurement of where you are and where you were displacement is a measurement of the change in position
Shii
Thanks a lot
Usman
I'm beginner in physics so I can't reason why v=u+at change to v2=u2+2as and vice versa
Usman
what is kinematics
praveen
kinematics is study of motion without considering the causes of the motion
Theo
The study of motion without considering the cause 0f it
Usman
why electrons close to the nucleus have less energy and why do electrons far from the nucleus have more energy
Theo
thank you frds
praveen
plz what is the third law of thermodynamics
third law of thermodynamics states that at 0k the particles will collalse its also known as death of universe it was framed at that time when it waa nt posible to reach 0k but it was proved wrong
bhat
I have not try that experiment but I think it will magnet....
Hey Rev. it will
Jeff
I do think so, it will
Chidera
yes it will
lasisi
If a magnet is in a pool of water, would it be able to have a magnetic field?.
yes Stella it would
Jeff
formula for electric current
Fokoua
what are you given?
Kudzy
what is current
Fokoua
I=q/t
saifullahi
Current is the flow of electric charge per unit time.
saifullahi
What are semi conductors
saifullahi
materials that allows charge to flow at varying conditions, temperature for instance.
Mokua
these are materials which have electrical conductivity greater than the insulators but less than metal, in these materials energy band Gap is very narrow as compared to insulators
Sunil
materials that allows charge to flow at varying conditions, temperature for instance.
Obasi
wao so awesome
Fokoua
At what point in the oscillation of beam will a body leave it?
Atambiri
what is gravitational force
what is meant by the term law