Derivation for energy conservation in SHM.WITH EACH AND EVERY CLEAR STEPS

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Solution

when the particle moving along a straight line with an acceleration which is always towards a fixed point on the line and whose magnitude is proportional to the distance from the fixed point, then the motion is called simple harmonic motion [SHM].
For example a simple harmonic oscillator consisting of a weight attached to one end of a spring and the other end is fixed to the wall. If the mass is displaced from the equilibrium position, the spring exerts a restoring elastic force that obeys Hooke's law.Since it is having a mass it also obey Newton second law of motion as shown below.

So comparing the two forces we get the angular frequency of the oscillation as below

So here Angular frequency , w= [ k/m]^ 1/2
Then the the kinetic energy [KE], potential energy [PE] and total energy [TE] can be computed as below

Hence in the absence of friction and other energy loss, the totalmechanical energy has a constant value .

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