# Energy calculation in Simple harmonic motion

Hello
why we use cosine and sine in simple harmonic motion?
why we use particularly cosine with potential energy and sine with kinetic energy of simple harmonic oscillator?
regards

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mfb
Mentor
The general solution for position and velocity is a sum of a cosine and a sine. If the starting conditions are not relevant, you can choose the origin of time (t=0) such that one term is a cosine and the other one is a sine.

thanks sir
but why we use these sine and cosine?
what these sine and cosine represent?

mfb
Mentor
What do you mean with "represent"? They are the solutions to the differential equations which describe the motion of the system.

ZapperZ
Staff Emeritus
thanks sir
but why we use these sine and cosine?
what these sine and cosine represent?
Unless you specifically know the gender of the person you are responding, please do not assume that all of us here are males. You are insulting the many female physicists, physics students, and participants in this forum.

Zz.

Mathematically, as mfb said, the summation of a sine and cosine is the solution to the differential equation that governs simple harmonic motion. For a mechanical oscillator, this equation is mx''+kx = 0, and the general solution is x(t) = c1*cos(wt+p)+c2*sin(wt+p). If you want a more intuitive understanding, you must realize that sine and cosine provide us a way to quantify circular motion. Conveniently, we can express the movement of an oscillator as a function of cosines and sines that have a constant angular frequency.

Here is a good animation that will hopefully make things more clear: In that animation, picture the oscillator as being the up and down movement of that rod as it goes around the circle, which is simply the sine wave.

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Or you can say that the very definition of a Simple Harmonic Motion is that where the position, velocity, acceleration of the particle vary sinusoidally with time

Unless you specifically know the gender of the person you are responding, please do not assume that all of us here are males. You are insulting the many female physicists, physics students, and participants in this forum.

Zz.
Yes ma'am.

Unless you specifically know the gender of the person you are responding, please do not assume that all of us here are males. You are insulting the many female physicists, physics students, and participants in this forum.

Zz.
!!!!!! are you going to check every post for such 'serious' infringments ??
Assuming someone is male is not an insult to anyone unless someone wants to take it as an insult.
Come on look at the questions!!!!

Back to the issue....Potential energy is to do with force x displacement...if the expression for displacement involves Cos(ωt) (x = ACosωt ?) then it is no surprise.
Kinetic energy is to do with velocity and v = dx/dt so if x is a Cos function then v is a Sin function v = ωASin(ωt).

sophiecentaur
Gold Member
Unless you specifically know the gender of the person you are responding, please do not assume that all of us here are males. You are insulting the many female physicists, physics students, and participants in this forum.

Zz.
Would "Oh wise one" be appropriate?
This gender thing is, indeed, a general problem. The word "they" tends do be used for "he or she" and it is a bit gramatically unsatisfactory. I wonder whether we could have a ruling on this from the boffins in PF Towers?

sophiecentaur