Shm , calculation of amplitude of spring mass system

[Shivang kohlii]

Homework Statement

In A spring mass system , the spring stretches 2 cm from its 's frelength when a force of 10 N is applied . This spring is stretched 10 cm from it's free length , when a body of mass m = 2 kg is attached to it and released from rest at time t = 0 . Find the A) force constant of spring
B) amplitude of vibration

Homework Equations

kx = mg
X is the extension from mean position
Elastic potential energy = 1/2 kx^2

The Attempt at a Solution

A) k = 10 / 0.02 = 500N/m
B ) considering the mean position as the point of extension of spring due to mass m ,
Energy at mean position ( X= 0 )= energy at maximum displacement( X= Amplitude , A )

Kinetic energy at mean position
=1/2 m×ω^2× A
= (1/2) × 2 × (k/m) × A
= 250 × A Joules

Gravitational potential energy at mean position ( taking base as the extreme position ) = mgA =
2 × 10 × A = 20A

Spring potential energy at mean position = 0

ii) kinetic energy at extreme position = gravitational potential energy at extreme position = 0
Elastic potential energy at extreme position = 1/2 × 500 × (A)^2 = 250 ×A^2

Equating the energies :
A = 1.8 but answer is 0.06 m

I am wondering if I made some mistake in the initial energy ? If yes , please point the mistake out with correction.. any help will be much appreciated ! Thanks

 

[haruspex]

Spring potential energy at mean position = 0

The spring is not relaxed at equilibrium.

There is a much easier way. Just find the extension at equilibrium.

 

[Shivang kohlii]

The spring is not relaxed at equilibrium.

There is a much easier way. Just find the extension at equilibrium.

Yes I know the method through extension at equilibrium ... But I want to check out the method of COE for future purposes...
Can u please just tell me the equation that u would use and the two points that u would consider for solving through COE ?

 

[haruspex]

Can u please just tell me the equation that u would use and the two points that u would consider for solving through COE ?

Try again for the spring PE at equilibrium.