Answer :
The displacement of SHO at which the energy half kinetic and half potential is
A/√2
How to calculate displacement if KE and PE are half?
In case of, no frictional forces on a simple harmonic oscillator the kinetic and potential energies of the simple harmonic oscillator are transformed alternatively, but total mechanical energy is conserved.
At equilibrium position the total energy of the simple harmonic oscillator is equal to kinetic energy
The potential energy U of a simple harmonic oscillator at a given displacement x from the equilibrium position is given as follows:
U=1/2kx²
Here, k is the spring constant.
The kinetic energy K of a simple harmonic oscillator at a given displacement is given as follows:
K=1/2mv²
Here, m is the mass attached to the spring, and v is the speed of the mass at a given displacement.
The total mechanical energy E of a system is equal to the sum of potential energy U and kinetic energy K.
E=U+K
Substitute,
E=1/2kx²+1/2mv²
At extreme position, the displacement x is equal to amplitude A, and speed of oscillator is equal to zero.
Substitute
E=1/2kA²+12m(0)²
E=1/2kA²
Now, if the total mechanical energy is proportional to square of the amplitude of the simple harmonic oscillator, then, the potential energy U of a simple harmonic oscillator at a given displacement x is,
U=1/2kx²
Equate half of total mechanical energy and potential to find the displacement.
E/2=U
Substitute
1/2(1/2kA²) =1/2kx²
A²/2=x²
x=A/√2
The displacement of SHO at which the energy half kinetic and half potential is
A/√2
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