At what displacement of a sho is the energy half kinetic and half potential? what fraction of the total energy of a sho is kinetic and what fraction potential when the displacement is one third the amplitude

The fraction of kinetic energy and  fraction of Kinetic energy and Potential energy  is mathematically given as

a)  

b)  F=8/9

c) Fp=1/9

Generally the equation for the Kinetic energy and Potential energy as is mathematically given as

KE = 1/2mw^2(A^2 - x^2)

and

PE = 1/2mw^2x^2

With

K.E and P.E equal we have

b) for when  x = A/3

Generally the equation for the Kinetic energy is mathematically given as

KE = 1/2mw^2(A^2 - x^2)

herefore, the fraction of  Kinetic energy  is

f = 8/9

Therefore,  fraction of PE is

Fp=1-f

Fp=1-(8/9)

Fp=1/9

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As we know that KE and PE is same at a given position

so we will have as a function of position given as

also the PE is given as function of position as

now it is given that

KE = PE

now we will have

so the position is 0.707 times of amplitude when KE and PE will be same

Part b)

KE of SHO at x = A/3

we can use the formula

now to find the fraction of kinetic energy

now since total energy is sum of KE and PE

so fraction of PE at the same position will be

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