Friday, February 28, 2014

WCPE paper is out

World Conference Physics Education (WCPE) paper is out!

page 413-438 of the WCPE proceedings.  http://www.wcpe2012.org/WCPE_2012_proceeding_book/7-WCPE2012(353-498).pdf
mirror: https://www.dropbox.com/s/rqu7c04uvbxzj9n/7-WCPE2012%28353-498%29.pdf



page 413-438 of the WCPE proceedings.  http://www.wcpe2012.org/WCPE_2012_proceeding_book/7-WCPE2012(353-498).pdf
mirror: https://www.dropbox.com/s/rqu7c04uvbxzj9n/7-WCPE2012%28353-498%29.pdf

Table of Lead school, computer model etc page 413-438 of the WCPE proceedings.  http://www.wcpe2012.org/WCPE_2012_proceeding_book/7-WCPE2012(353-498).pdf
mirror: https://www.dropbox.com/s/rqu7c04uvbxzj9n/7-WCPE2012%28353-498%29.pdf

Normalized (Hake's) Gain's Regression Analysis: page 413-438 of the WCPE proceedings.  http://www.wcpe2012.org/WCPE_2012_proceeding_book/7-WCPE2012(353-498).pdf
mirror: https://www.dropbox.com/s/rqu7c04uvbxzj9n/7-WCPE2012%28353-498%29.pdf

page 413-438 of the WCPE proceedings.

https://www.dropbox.com/s/rqu7c04uvbxzj9n/7-WCPE2012%28353-498%29.pdf



Permission Granted to reproduce and share

Dear Loo Kang Lawrence Wee,
We are happy to announce that the book of proceedings is now ready and can be downloaded from the conference website:wcpe2012.org
The conference website will be maintained for two more years. Therefore, for your records please download the book and feel free to share and publish it on your personal or institutional websites.
We wish that this most voluminous book on physics education will set a tradition for the next WCPE's and thank all those who contributed to it.

Best regards,
Prof. Dr. M. Fatih Tasar
Chair of the Organizing Committee of the WCPE 2012

Wednesday, February 26, 2014

Interactive learning resources

Innergy award hq 2014. Thanks to @engrg1 for your leadership.
We also took a photo to remember selfie days.
Innergy award hq 2014. Thanks to @engrg1 for your leadership.
LEFT to RIGHT: @lookang ( Wee Loo Kang Lawrence), @engrg1 (Lye Sze Yee ),  Cynthia Seto and Khoo Ghee Han
test

commendation.

eduLabAST 2014 nice project highlights

nice project highlights,a photo taken by charles surin
a photo taken by Charles surin. thanks man :)

Monday, February 24, 2014

EJSS circle motion to SHM model

Update 10 July 2015

practice of mathematical modeling added

simple play and observe
http://weelookang.blogspot.sg/2015/01/shm-chapter-03.html
Run: Link1, Link2
Download and Unzip: Link1, Link2
Source File: Link1, Link2
author: lookang
author of EJSS 5.0 Francisco Esquembre

select a model (TEAL) that can describe simulated data (BLUE), note that Y = 2*sin(t) is too big compared to ϑ1 BLUE's motion, which you can select from drop-down menu and modify through input field
http://weelookang.blogspot.sg/2015/01/shm-chapter-03.html
Run: Link1Link2
Download and Unzip: Link1Link2
Source File: Link1Link2
author: lookang
author of EJSS 5.0 Francisco Esquembre

select a model (TEAL) that can describe simulated data (BLUE), note that Y = 0.97*sin(t) fits well  to ϑ1 BLUE's motion, 
http://weelookang.blogspot.sg/2015/01/shm-chapter-03.html
Run: Link1Link2
Download and Unzip: Link1Link2
Source File: Link1Link2
author: lookang
author of EJSS 5.0 Francisco Esquembre

Play and observe that the model (TEAL) can describe simulated data (BLUE)note that Y = 0.97*sin(t) for another period of extended play.
http://weelookang.blogspot.sg/2015/01/shm-chapter-03.html
Run: Link1Link2
Download and Unzip: Link1Link2
Source File: Link1Link2
author: lookang
author of EJSS 5.0 Francisco Esquembre

select a model (TEAL) that can describe simulated data (MAGENTA)note that Y = -0.88*cos(t) fits well to to ϑ2 (MAGENTA)'s motion, which you can select from drop-down menu and modify through input field
http://weelookang.blogspot.sg/2015/01/shm-chapter-03.html
Run: Link1Link2
Download and Unzip: Link1Link2
Source File: Link1Link2
author: lookang
author of EJSS 5.0 Francisco Esquembre




EJSS circle motion to SHM model
EJSS simple harmonic motion to circular motion model with phase difference
based on models and ideas by
http://weelookang.blogspot.sg/2014/02/ejss-pendulum-model.html
EJSS circle motion to SHM model
https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_SHMcircle/SHMcircle_Simulation.html
source code: https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_src_SHMcircle.zip
experimental http://phonegap.com/ android app: https://dl.dropboxusercontent.com/u/44365627/EJSScirclemotiontoSHMmodel-debug%20%281%29.apk
author: lookang
author of EJSS 5.0 Francisco Esquembre
  1. lookang http://weelookang.blogspot.sg/2012/07/ejs-open-source-phase-difference-java.html
  2. lookang http://weelookang.blogspot.sg/2013/02/ejs-open-source-vertical-spring-mass.html?q=vertical+spring

http://weelookang.blogspot.sg/2014/02/ejss-pendulum-model.html
EJSS circle motion to SHM model
https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_SHMcircle/SHMcircle_Simulation.html
source code: https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_src_SHMcircle.zip
experimental http://phonegap.com/ android app: https://dl.dropboxusercontent.com/u/44365627/EJSScirclemotiontoSHMmodel-debug%20%281%29.apk
author: lookang
author of EJSS 5.0 Francisco Esquembre

Description:

In physics, uniform circular motion describes the motion of a body traversing a circular path at constant speed. The distance of the body from the axis of rotation remains constant at all times. Though the body's speed is constant, its velocity is not constant: velocity, a vector quantity, depends on both the body's speed and its direction of travel. This changing velocity indicates the presence of an acceleration; this centripetal acceleration is of constant magnitude and directed at all times towards the axis of rotation. This acceleration is, in turn, is responsible for a centripetal force which is also constant in magnitude and directed towards the axis of rotation.


The equations that model the motion of the circular motion are:

this uniform $ \omega_{1} $ =$ \omega_{2} $ circular motion model assumes

$ \frac{\delta \theta_{1}}{\delta t} = \omega_{1} $

$ \frac{\delta \theta_{2}}{\delta t} = \omega_{2} $


where the terms


$ \theta_{1} $ and $ \theta_{2} $ represents the angle of rotation in uniform circular motion

$ \omega_{1} $ and $ \omega_{2} $ are constants equal to each other.
in circular motion, 

$ \theta_{1} = \omega_{1} t $

$ \theta_{2} = \omega_{2} t $

results in phase difference of



$ \phi =  \theta_{1} -  \theta_{2} $ when rotation is clockwise


$ \phi =  \theta_{2} -  \theta_{1} $ when rotation is anti-clockwise viewed from your perspective.

Simple harmonic motion can in some cases be considered to be the one-dimensional projection of uniform circular motion. If an object moves with angular speed $ \omega $ around a circle of radius $ A $ centered at the origin of the $ x-y $ plane, then its motion along each coordinate is simple harmonic motion with amplitude $ A $ and angular frequency $ \omega $.

The simplified equations that model the motion projection of circular motion = simple harmonic motion are:

if $ y_{1} = A_{1} cos (\omega_{1}t ) $

then $ y_{2} = A_{2} cos (\omega_{2}t - \phi) $

and

if $ x_{1} = -A_{1} sin(\omega_{1}t ) $

then $ x_{2} = A_{2} sin (\omega_{2}t + \phi) $


Friday, February 21, 2014

MOE Contact Teacher Digest Jan 2014 on page 16&17

Read about the Tips on Building a learning network online from @tucksoon , @shamsensei and @lookang !
Contact Teacher Digest Jan 2014 on page 16&17
Many thanks to MOE and http://www.moe.gov.sg/corporate/contactprint/index.htm, and http://www.tuberproductions.com/project/?project=8 and Challenge Magazine photo by John Heng.


Contact Teacher Digest Jan 2014 on page 16&17. Building a learning network online by @tucksoon , @shamsensei and @lookang. to be available here http://www.moe.gov.sg/corporate/contactprint/index.htm

Contact Teacher Digest Jan 2014 on page 16&17. Building a learning network online by @tucksoon , @shamsensei and @lookang. to be available here http://www.moe.gov.sg/corporate/contactprint/index.htm
Thanks to Mr Sham ‏@shamsensei Feb 19
@ashley Contact. Mentioned #efiesta as well. And our gd friend @lookang posing hahaha. Good coverage #edsg pic.twitter.com/WrzqDm1Vdq

Thanks to Mr Sham ‏@shamsensei Feb 19
@ashley Contact. Mentioned #efiesta as well. And our gd friend @lookang posing hahaha. Good coverage #edsg pic.twitter.com/WrzqDm1Vdq

Tuesday, February 18, 2014

EJSS pendulum model

EJSS SHM pendulum model with t vs $ \theta $ graph
EJSS simple harmonic motion pendulum model with t vs $ \theta $ graph
based on models and ideas by
http://weelookang.blogspot.sg/2014/02/ejss-pendulum-model.html
EJSS SHM pendulum model with t vs $ \theta $ graph
https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_SHMxvapendulumv2/SHMxvapendulumv2_Simulation.html
source code: https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_src_SHMxvapendulumv2.zip
author: lookang
author of EJSS 5.0 Francisco Esquembre
  1. lookang http://weelookang.blogspot.sg/2014/02/ejss-shm-model-with-vs-x-and-v-vs-x.html
  2. lookang http://weelookang.blogspot.sg/2010/06/ejs-open-source-simple-harmonic-motion.html?q=SHM
  3. lookang http://weelookang.blogspot.sg/2013/02/ejs-open-source-vertical-spring-mass.html?q=vertical+spring
  4. lookang http://weelookang.blogspot.sg/2010/06/physical-quantities-and-units.html?q=pendulum
  5. Wolfgang Christian and Francisco Esquembre http://www.opensourcephysics.org/items/detail.cfm?ID=13103



http://weelookang.blogspot.sg/2014/02/ejss-pendulum-model.html
EJSS SHM pendulum model with t vs $ \theta $ graph
https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_SHMxvapendulumv2/SHMxvapendulumv2_Simulation.html
source code: https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_src_SHMxvapendulumv2.zip
author: lookang
author of EJSS 5.0 Francisco Esquembre
File:Simple gravity pendulum.svg
commons terms associated with pendulum http://en.wikipedia.org/wiki/File:Simple_gravity_pendulum.svg#file

Assumption of this simple pendulum model:


  1. The rod or cord on which the bob swings is massless, inextensible and always remains taut;
  2. The bob is a point mass;
  3. Motion occurs only in two dimensions, i.e. the bob does not trace an ellipse but an arc.
  4. The motion does not lose energy to friction or air resistance.

Assumptions of SHM comparable to pendulum:

  1. Motion approximates SHM when the angle $ \theta$ of oscillation is small, where $ sin \theta \approx \theta $

The equations that model the motion of the pendulum system are:

this model assumes

$ \frac{\delta \theta}{\delta t} = \omega $


$ \frac{\delta \omega}{\delta t} = -\frac{g}{L}( sin \theta)  $

where the terms

$ L $ represents the fixed length of the pendulum 

$  g $ represents the gravity force component as a result of Earth's pull.



If the motion starts to the positive amplitude position:

$ \theta = 5 degree $
$ \theta = 10 degree $
$ \theta = 15 degree $
$ \theta = 20 degree $
$ \theta = 25 degree $
$ \theta = 30 degree $
$ \theta = 40 degree $
$ \theta = 50 degree $
$ \theta = 60 degree $
$ \theta = 70 degree $
$ \theta = 80 degree $
$ \theta = 90 degree $

General Rule of Thumb

Thus, in general, the pendulum's swing motion is approximately a simple harmonic motion only when the Motion's angle $ \theta$ of oscillation is small, where $ sin \theta \approx \theta $, rule of thumb is about 5 degree!


EJSS vertical spring mass model

EJSS SHM vertical spring mass model with y vs t, v vs t and a vs t graph
EJSS simple harmonic motion vertical spring mass model with y vs t, v vs t and a vs t graph
based on models and ideas by

  1. lookang http://weelookang.blogspot.sg/2014/02/ejss-shm-model-with-vs-x-and-v-vs-x.html
  2. lookang http://weelookang.blogspot.sg/2010/06/ejs-open-source-simple-harmonic-motion.html?q=SHM
  3. lookang http://weelookang.blogspot.sg/2013/02/ejs-open-source-vertical-spring-mass.html?q=vertical+spring
  4. Wolfgang Christian and Francisco Esquembre http://www.opensourcephysics.org/items/detail.cfm?ID=13103
http://weelookang.blogspot.com/2014/02/ejss-vertical-spring-mass-model.html
EJSS SHM vertical spring mass model with y vs t, v vs t and a vs t graph
https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_SHMxvavertical/SHMxvavertical_Simulation.html
source: https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_src_SHMxvavertical.zip
author: lookang
author of EJSS 5.0 Francisco Esquembre

Assumption:

Motion approximates SHM when the spring does not exceed limit of proportionality during oscillations.

http://weelookang.blogspot.com/2014/02/ejss-vertical-spring-mass-model.html
EJSS SHM vertical spring mass model with y vs t, v vs t and a vs t graph
https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_SHMxvavertical/SHMxvavertical_Simulation.html
source: https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_src_SHMxvavertical.zip
author: lookang
author of EJSS 5.0 Francisco Esquembre




The equations that model the motion of the spring mass system are:

Mathematically, the restoring force $ F $ is given by 


$ F = - k y $

where $ F $  is the restoring elastic force exerted by the spring (in SI units: N), k is the spring constant (N·m−1), and y is the displacement from the equilibrium position (in m).

Thus, this model assumes 

$ \frac{\delta y}{\delta t} = v_{y} $


$ \frac{\delta v_{y}}{\delta t} = -\frac{k}{m}(y-l) - g  $

where the terms

$ -\frac{k}{m}(y-l) $ represents the restoring force component as a result of the spring extending and compressing.

$ - g $ represents the gravity force component as a result of Earth's pull.


What is SHM?

Simple harmonic motion is typified by the motion of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's Law. The motion is sinusoidal in time and demonstrates a single resonant frequency. In order for simple harmonic motion to take place, the net force of the object at the end of the pendulum must be proportional to the displacement. In other words, oscillations are periodic variations in the value of a physical quantity about a central or equilibrium value.

Once the mass is displaced from its equilibrium position, it experiences a net restoring force. As a result, it accelerates and starts going back to the equilibrium position. When the mass moves closer to the equilibrium position, the restoring force decreases. At the equilibrium position, the net restoring force vanishes. However, at y = 0, the mass has momentum because of the impulse that the restoring force has imparted. Therefore, the mass continues past the equilibrium position, compressing the spring. A net restoring force then tends to slow it down, until its velocity reaches zero, whereby it will attempt to reach equilibrium position again.


If motion starts at the equilibrium position and starts to move to the positive direction solutions to the defining equation are:

not possible, as it is in equilibrium.


If the motion starts to the negative amplitude position:

$ y = - y_{o} cos ( \omega t ) = y_{o} sin ( \omega t -  \frac{\pi }{2} )$
$ y = - y_{o} cos ( \omega t ) = y_{o} sin ( \omega t -  \frac{\pi }{2} )$


$ v = y_{o} \omega sin ( \omega t ) = y_{o} \omega cos ( \omega t -  \frac{\pi }{2} )$
$ v = y_{o} \omega sin ( \omega t ) = y_{o} \omega cos ( \omega t -  \frac{\pi }{2} )$


$ a =  y_{o} \omega^{2} cos ( \omega t ) = - y_{o} \omega^{2} sin ( \omega t -  \frac{\pi }{2} )$
$ a =  y_{o} \omega^{2} cos ( \omega t ) = - y_{o} \omega^{2} sin ( \omega t -  \frac{\pi }{2} )$




therefore , in general:

$ y = y_{o} sin ( \omega t - \phi ) $

where

$ \phi = \pi/2 $ for a starting position of  $ y = - y_{o}$
$ \phi = - \pi/2 $ for a starting position of  $ y = y_{o}$


$ v = y_{o} \omega cos ( \omega t - \phi ) $

$ a = - y_{o} \omega^{2} sin ( \omega t - \phi ) $




Friday, February 14, 2014

EJSS SHM e vs x

EJSS SHM model with e vs x graph
EJSS simple harmonic motion model with energy vs x graph showing elastic potential, kinetic and total energy
based on models and ideas by
http://weelookang.blogspot.sg/2014/02/ejss-shm-e-vs-x.html
EJSS simple harmonic motion model with energy vs t graph showing elastic potential, kinetic and total energy
https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_SHMet/SHMet_Simulation.html
source: https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_src_SHMex.zip
author: lookang
author of EJSS 5.0 Francisco Esquembre
  1. lookang http://weelookang.blogspot.sg/2010/06/ejs-open-source-simple-harmonic-motion.html?q=SHM
  2. Wolfgang Christian and Francisco Esquembre http://www.opensourcephysics.org/items/detail.cfm?ID=13103
http://weelookang.blogspot.sg/2014/02/ejss-shm-e-vs-x.html
EJSS simple harmonic motion model with energy vs t graph showing elastic potential, kinetic and total energy
https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_SHMet/SHMet_Simulation.html
source: https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_src_SHMex.zip
author: lookang
author of EJSS 5.0 Francisco Esquembre

notes:

http://weelookang.blogspot.sg/2014/02/ejss-shm-e-vs-x.html
EJSS simple harmonic motion model with energy vs t graph showing elastic potential, kinetic and total energy
https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_SHMet/SHMet_Simulation.html
source: https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_src_SHMex.zip
author: lookang
author of EJSS 5.0 Francisco Esquembre


The equations that model the motion of the spring mass system are:

Mathematically, the restoring force $ F $ is given by 


$ F = - k x $

where $ F $  is the restoring elastic force exerted by the spring (in SI units: N), k is the spring constant (N·m−1), and x is the displacement from the equilibrium position (in m).

Thus, this model assumes 

$ \frac{\delta x}{\delta t} = v_{x} $


$ \frac{\delta v_{x}}{\delta t} = -\frac{k}{m}(x-l) - \frac{bv_{x}}{m} + \frac{A sin(2 \pi f t)}{m} $

where the terms

$ -\frac{k}{m}(x-l) $ represents the restoring force component as a result of the spring extending and compressing.

$ - \frac{bv_{x}}{m}$ represents the damping force component as a result of drag retarding the mass's motion.

$ + \frac{A sin(2 \pi f t)}{m} $ represents the driving force component as a result of a external periodic force acting the mass $ m $.

What is SHM?

Simple harmonic motion is typified by the motion of a mass on a spring when it is subject to the linear elastic restoring force given by Hooke's Law. The motion is sinusoidal in time and demonstrates a single resonant frequency. In order for simple harmonic motion to take place, the net force of the object at the end of the pendulum must be proportional to the displacement. In other words, oscillations are periodic variations in the value of a physical quantity about a central or equilibrium value.

Once the mass is displaced from its equilibrium position, it experiences a net restoring force. As a result, it accelerates and starts going back to the equilibrium position. When the mass moves closer to the equilibrium position, the restoring force decreases. At the equilibrium position, the net restoring force vanishes. However, at x = 0, the mass has momentum because of the impulse that the restoring force has imparted. Therefore, the mass continues past the equilibrium position, compressing the spring. A net restoring force then tends to slow it down, until its velocity reaches zero, whereby it will attempt to reach equilibrium position again.

As long as the system has no energy loss, the mass will continue to oscillate. Thus, simple harmonic motion is a type of periodic motion.


Energy of SHM:

Consider a spring of spring constant k connected to a mass m as shown.  When the mass is displaced from its equilibrium position by a distance xo and released, it will oscillate backwards and forwards

At the time of release, the energy of the system will consist totally of the spring’s potential energy. As the spring pulls the mass toward the equilibrium position, the potential energy is transformed into kinetic energy until at the equilibrium position the kinetic energy will reach a maximum. After the mass crosses the equilibrium position, the spring is compressed and the EP is converted to EK. Maximum PE occurs at maximum compression of spring.

The constant exchange between potential energy EP and kinetic energy EK is essential in producing oscillations. The total energy ET, which is the summation of kinetic and potential energy, will always add up to a constant value as shown.

Variation with time of energy in simple harmonic motion are as follows:

If motion starts at the equilibrium position and starts to move to the positive direction solutions to the defining equation are:

$ x = x_{o} sin ( \omega t ) $
motion starts at the equilibrium position and starts to move to the positive direction, defining equation follows  $ x = x_{o} sin ( \omega t ) $


$ v = x_{o} \omega cos ( \omega t ) $

 motion starts at the equilibrium position and starts to move to the positive direction, defining equation follows $ v = x_{o} \omega cos ( \omega t ) $



$ a = - x_{o} \omega^{2} sin ( \omega t ) $
motion starts at the equilibrium position and starts to move to the positive direction, defining equation follows $ a = - x_{o} \omega^{2} sin ( \omega t ) $



$ KE = \frac{1}{2}mv^{2} =  \frac{1}{2} m  \omega^{2} x_{o}^{2} cos^{2} ( \omega t ) $

$ PE = \frac{1}{2}kx^{2} =  \frac{1}{2} m  \omega^{2} x_{o}^{2} sin^{2} ( \omega t ) $

$ TE = KE + PE =   \frac{1}{2} m  \omega^{2} x_{o}^{2}  $
$ KE = \frac{1}{2}mv^{2} =  \frac{1}{2} m  \omega^{2} x_{o}^{2} cos^{2} ( \omega t ) $

$ PE = \frac{1}{2}kx^{2} =  \frac{1}{2} m  \omega^{2} x_{o}^{2} sin^{2} ( \omega t ) $

$ TE = KE + PE =   \frac{1}{2} m  \omega^{2} x_{o}^{2}  $



$ KE = \frac{1}{2}mv^{2} =  \frac{1}{2} m  \omega^{2} ( x_{o}^{2} -x^{2}) $

$ PE = \frac{1}{2}kx^{2} =  \frac{1}{2} m  \omega^{2} x^{2}  $

$ TE = KE + PE =   \frac{1}{2} m  \omega^{2} x_{o}^{2}  $
$ KE = \frac{1}{2}mv^{2} =  \frac{1}{2} m  \omega^{2} ( x_{o}^{2} -x^{2}) $
$ PE = \frac{1}{2}kx^{2} =  \frac{1}{2} m  \omega^{2} x^{2}  $
$ TE = KE + PE =   \frac{1}{2} m  \omega^{2} x_{o}^{2}  $





If the motion starts to the negative amplitude position:


$ x = - x_{o} cos ( \omega t ) = x_{o} sin ( \omega t -  \frac{\pi }{2} )$

motion starts at the negative position and starts to move to the positive direction, defining equation follows $ x = - x_{o} cos ( \omega t ) = x_{o} sin ( \omega t - \frac{\pi }{2} )$

$ v = x_{o} \omega sin ( \omega t ) = x_{o} \omega cos ( \omega t -  \frac{\pi }{2} )$

motion starts at the negative position and starts to move to the positive direction, defining equation follows $ v = x_{o} \omega sin ( \omega t ) = x_{o} \omega cos ( \omega t - \frac{\pi }{2} )$


$ a =  x_{o} \omega^{2} cos ( \omega t ) = - x_{o} \omega^{2} sin ( \omega t -  \frac{\pi }{2} )$

motion starts at the negative position and starts to move to the positive direction, defining equation follows $ a = x_{o} \omega^{2} cos ( \omega t ) = - x_{o} \omega^{2} sin ( \omega t - \frac{\pi }{2} )$


$ KE = \frac{1}{2}mv^{2} =  \frac{1}{2} m  \omega^{2} x_{o}^{2} sin^{2} ( \omega t ) $

$ PE = \frac{1}{2}kx^{2} =  \frac{1}{2} m  \omega^{2} x_{o}^{2} cos^{2} ( \omega t ) $

$ TE = KE + PE =   \frac{1}{2} m  \omega^{2} x_{o}^{2}  $

$ KE = \frac{1}{2}mv^{2} =  \frac{1}{2} m  \omega^{2} x_{o}^{2} sin^{2} ( \omega t ) $

$ PE = \frac{1}{2}kx^{2} =  \frac{1}{2} m  \omega^{2} x_{o}^{2} cos^{2} ( \omega t ) $

$ TE = KE + PE =   \frac{1}{2} m  \omega^{2} x_{o}^{2}  $




therefore , in general:

$ x = x_{o} sin ( \omega t - \phi ) $


$ v = x_{o} \omega cos ( \omega t - \phi ) $

$ a = - x_{o} \omega^{2} sin ( \omega t - \phi ) $


$ KE = \frac{1}{2}mv^{2} =  \frac{1}{2} m  \omega^{2} x_{o}^{2} cos^{2} ( \omega t - \phi) $

$ PE = \frac{1}{2}kx^{2} =  \frac{1}{2} m  \omega^{2} x_{o}^{2} sin^{2} ( \omega t - \phi) $

$ TE = KE + PE =   \frac{1}{2} m  \omega^{2} x_{o}^{2}  $

$ KE = \frac{1}{2}mv^{2} =  \frac{1}{2} m  \omega^{2} ( x_{o}^{2} -x^{2}) $

$ PE = \frac{1}{2}kx^{2} =  \frac{1}{2} m  \omega^{2} x^{2}  $

$ TE = KE + PE =   \frac{1}{2} m  \omega^{2} x_{o}^{2}  $