I am currently self-studying introductory physics and have come across an anomaly while attempting to derive the equation for the period of simple harmonic oscillations on a spring. This is how I did it; let me know where I went wrong.
Suppose there is a block of mass m at rest on a frictionless horizontal surface. Onto the side of this block is attached a spring (which obeys Hooke's law) that is also attached to a fixed wall. The spring is compressed a distance x and released, leading to SHM.
At the moment that the spring is compressed the full distance x, the magnitude of the force on the block is given by F = kx. From the time that the block is fully compressed to the time that it recovers to the equilibrium position, the average force exerted on it by the spring is 1/2kx.
Via Newton's second law, I calculated the average acceleration of the block during this same time frame, a = kx/2m.
Then I used kinematics to find the time it took for the block to complete this journey:
x-xo = vxot + 1/2at^2, x = 1/2(kx/2m)t^2, t^2 = sqrt(4m/k), t = 2sqrt(m/k)
Since this time frame is one-fourth of the total period, the period of the motion would be 8sqrt(m/k).
I know the real period is 2πsqrt(m/k), so what went wrong?
I know calculus but would prefer an algebra-based explanation.