Simple harmonic motion equations derivation?

[hey123a]

Well I was going through class lecture notes and my professor wrote this

When x = A(the maximum value), v=0: E=1/2kA^2
When v = wA, x=0: E=1/2mw^2A^2

where w = omega, A = amplitude, k = spring constant, m = mass, v = velocity

and apparently both equations are equal, i would like to know how and why
please and thank you

 

[Tanya Sharma]

If there is no friction or dissipation,kinetic and potential energy are alternately transformed into each other in SHM,but the total mechanical energy E=K+U is conserved.

Consider a spring with one end attached to the wall and other end having a block,going through its cycle of oscillation.

Total energy E = (1/2)mv² + (1/2)kx²

At any position x, E = (1/2)mv_x² + (1/2)kx²

At the extreme position ,i.e at x=A the block has only potential energy ,so E = (1/2)kA²

At the equilibrium position ,at x=0 the block has only kinetic energy ,so E = (1/2)mv_max²

Now we know,v_max = ωA ,so at x=0 E=(1/2)mω²A²