# 15.2 Energy in simple harmonic motion  (Page 2/8)

 Page 2 / 8

Consider [link] , which shows the energy at specific points on the periodic motion. While staying constant, the energy oscillates between the kinetic energy of the block and the potential energy stored in the spring:

${E}_{\text{Total}}=U+K=\frac{1}{2}k{x}^{2}+\frac{1}{2}m{v}^{2}.$

The motion of the block on a spring in SHM is defined by the position $x\left(t\right)=A\text{cos}\left(\omega t+\varphi \right)$ with a velocity of $v\left(t\right)=\text{−}A\omega \text{sin}\left(\omega t+\varphi \right)$ . Using these equations, the trigonometric identity ${\text{cos}}^{2}\theta +{\text{sin}}^{2}\theta =1$ and $\omega =\sqrt{\frac{k}{m}}$ , we can find the total energy of the system:

$\begin{array}{cc}\hfill {E}_{\text{Total}}& =\frac{1}{2}k{A}^{2}{\text{cos}}^{2}\left(\omega t+\varphi \right)+\frac{1}{2}m{A}^{2}{\omega }^{2}{\text{sin}}^{2}\left(\omega t+\varphi \right)\hfill \\ & =\frac{1}{2}k{A}^{2}{\text{cos}}^{2}\left(\omega t+\varphi \right)+\frac{1}{2}m{A}^{2}\left(\frac{k}{m}\right){\text{sin}}^{2}\left(\omega t+\varphi \right)\hfill \\ & =\frac{1}{2}k{A}^{2}{\text{cos}}^{2}\left(\omega t+\varphi \right)+\frac{1}{2}k{A}^{2}{\text{sin}}^{2}\left(\omega t+\varphi \right)\hfill \\ & =\frac{1}{2}k{A}^{2}\left({\text{cos}}^{2}\left(\omega t+\varphi \right)+{\text{sin}}^{2}\left(\omega t+\varphi \right)\right)\hfill \\ & =\frac{1}{2}k{A}^{2}.\hfill \end{array}$

The total energy of the system of a block and a spring is equal to the sum of the potential energy stored in the spring plus the kinetic energy of the block and is proportional to the square of the amplitude ${E}_{\text{Total}}=\left(1\text{/}2\right)k{A}^{2}.$ The total energy of the system is constant.

A closer look at the energy of the system shows that the kinetic energy oscillates like a sine-squared function, while the potential energy oscillates like a cosine-squared function. However, the total energy for the system is constant and is proportional to the amplitude squared. [link] shows a plot of the potential, kinetic, and total energies of the block and spring system as a function of time. Also plotted are the position and velocity as a function of time. Before time $t=0.0\phantom{\rule{0.2em}{0ex}}\text{s,}$ the block is attached to the spring and placed at the equilibrium position. Work is done on the block by applying an external force, pulling it out to a position of $x=+A$ . The system now has potential energy stored in the spring. At time $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s,}$ the position of the block is equal to the amplitude, the potential energy stored in the spring is equal to $U=\frac{1}{2}k{A}^{2}$ , and the force on the block is maximum and points in the negative x -direction $\left({F}_{S}=\text{−}kA\right)$ . The velocity and kinetic energy of the block are zero at time $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}\text{.}$ At time $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s,}$ the block is released from rest.

#### Questions & Answers

who discover periodic table?
it wasn't discovered , it was made.and the person who made it was dmitri mendeleev. dobreinier and newland gave their laws before dmitri related periodic table but wasn't successful in their work
Ritik
Nope, numerous number of scientist had actually contributed in the making of periodic table. Dmitri Mendeleev succeeded making all the elements into the right order in accordance to their atomic number.
Dame
what is the Greek name for calcium
different types of wave
longitudinal and transverse waves
Ravindra
a gun is kept in the state that it cannot move anywhere and the bullet is fired. Then what is the effect on the velocity of bullet and KE of gun ?
want is meant by the term solar system
it refers to the sun and all heavenly bodies revolving around it.
Danie
excatly...for sure
Arzoodan
in addition to Danie's, a solar system is a collection of planets and their moons, asteroids, and other objects bound together by the Star's gravitational force directly or indirectly.
Galiwango
what is meant by total internal reflection
what iw meant by total internal reflection
Akshay
Lorentz force?
jyotirmayee
study fibre optics. .you will get total internal reflection
Siddhansh
wha
jyotirmayee
what is Lorentz force?
jyotirmayee
a ray of light traveling at an angle of incidence greater than critical angle from denser to rarer medium is totally reflected back into the denser medium is called total internal reflaction
Manoj
motion in strat line where is this chapter
motion in straight line is kinematic's part
Ritik
yea
Manoj
this defination isn't correct
Arzoodan
motion in one dimension
Anil
what is Lorentz force?
what is maxwell electromagnetic law?
jyotirmayee
at what angle should the two forces 2p and root 2p acts so that the resultant force is p root 10
what answer fir this
Akshay
what's the working difference between a dynamo and a pump?
a dynamo is basically a dc generator while pump is usually equipped with a motor
vedanth
a dynamo converts mechanical energy to electrical while a pump is opposite to that
vedanth
nice
Piyali
okay
Friday
why sea water looks bluish?
cuz the sky is blue...
Mehdi
you see the reflexion of the "blue" sky in the water
Mehdi
somewhere sea water turns green why?
Piyali
never seen bro... are u sure ?
Mehdi
ur answer was correct but due to the presence of phytoplankton color can be changed near the shore
Piyali
waaaww... you re awesome
Mehdi
because of the reflection of the sky
Friday
rays coming from the sun consist of all 7 colours ie.VIBGYOR. when the ray strikes surface of water,all colors gets absorbed by it except blue which gets reflected by it.so we find the sea water appearing bluish
Ritik
how can someone identify sea water from rain water
Oniyide
i think it's not possible as because rainwater consists of water from all kind of water bodies ie.lakes,seas etc but u can predict if u have a sea nearby ur home or city
Ritik
i need solutions of unuversity pbysucs volume 1
me too
Nirupam
Help us if anyone knows
Nirupam
bring questions
john
actually if u wanted whole book solution then u should buy the solution book
jyotirmayee
from where
Nirupam
where do u live ,,,,,if u live in Delhi then at bellsarayeiii or from stationary store ,,,,chatarpurrr ,,,,,there is a popular books store,,,,,u have to buy from there
jyotirmayee
Kota Rajasthan
Nirupam
so strange,,,,,r u preparing for pmt,,,?
jyotirmayee
for IIT
Nirupam
then concern near book store
jyotirmayee
in Rajasthan
jyotirmayee
or ask questions here
jyotirmayee
thanks for your help
Nirupam
its fine,,most welcome
jyotirmayee
what is three dimensional coordinate system?
considering the change of vectors in all three dimensions of space
vedanth
direction co-sign of vector questions
opposite
Sonu
opposite mean
Vipin
what is the different between action and reaction?
action is external force. Reaction exists because of the external action. Reaction is mostly an internal force
Yoblaze
The difference is said in the word itself. There is no existance of reaction without an application of action. So action occur first then reaction. Reaction may produce in body itself or to other body i.e., it may be internal or external.
Amalesh
or we can say , reaction is the result of action
Ritik