Question

Derive the expration for enargy of simple harmonic motion​

Answer 1

Answer:

If an object is under oscil lating motion and at any in stan t the force acting on it is prop ortional to its displac ement from equ ilib rium position, then such an oscilla ting motion is called Simple har monic motion (SHM).

Mathematically F(x) = -k×x..(1)

where k is constant and negative sign indic ates direct ion of for ce and direc tion of displac ement are oppo site.

Equation (1) can be rewritten by using the definition of force = mass times acceleration

where x is displacement, xm is amplitude of SHM, ω is angular frequency;

ω= 2πf where f is the frequency of oscillation and Φ is phase constant.

hence total energy is constant

Answer 2

Answer:

Expression of energy of Simple Harmonic Motion :-

E = 1/ 2kA²

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Derivation :-

We know that,

Total Energy(E) = Kinetic Energy(KE) + Potential Energy (PE)

• KE = $\frac{1}{2}$mv²

= 1/ 2 m(Aω cos(ωt))² .  .  .  .  .  . [ ∵ v = Aω cos(ωt) ]

= 1/ 2 mA²ω² cos²(ωt)

• PE = $\frac{1}{2}$kx²

= 1/ 2 KA²sin²(ωt)   .  .  .   .  .   .  [ x = Asin (ωt)]

= 1/ 2 mω²A²sin²(ωt)  .  .  .  .   .  . [k = mω²]

=  1/ 2 mω²A²sin²(ωt)

• E = PE + KE

=  1/ 2 mω²A²sin²(ωt) + 1/ 2 mA²ω² cos²(ωt)

= 1/ 2 mω²A² (sin²(ωt) + cos²(ωt))

= 1/ 2 mω²A² (1)     .  .  .   . . [sin²(ωt) + cos²(ωt)]

= 1/ 2 kA²   .  .  .   . . [k = mω²]

Thus, E = 1/ 2 kA²