I'm haivng troulbe proving the deriviation of this formula on my own...

Question

could you please help me derive the proof of the formula or maybe show me one thansk...

I need to prove this formula

period of a mass spring system in simple harmonic motion

T = 2 pi square root of (m/k)
period = 2 pi X square root of (mass divided by spring constant)

thanks

Answer 1

F = m a
F = -k x for spring of constant k in N/m
so
m a = - k x
now assume motion of form x = A sin (2 pi f t)
then
dx/dt = velocity = A (2 pi f) cos 2 pi f t)
then
d velocity/dt = a = -A (2 pi f)^2 sin (2 pi f t)
or in other words
a = - x (2 pi f)^2
so
- k x = m [ - x (2 pi f)^2 ]
or
(2 pi f)^2 = k/m
so
2 pi f = sqrt (k/m)
but T = 1/f
so
T = 2 pi sqrt (m/k)