# Period of a mass oscillating on a spring

#### jgens

Gold Member
1. The problem statement, all variables and given/known data

The period of a mass m oscillating on a hookian spring of negligible mass.

2. Relevant equations

F = -kx

3. The attempt at a solution

I'm using x in place of ∆x for convienance.

F = -kx

dv/dt = -kx/m

∫vdv = -k/m∫xdx

v^2 = C - k(x^2)/m: Placing the restraint that if v = 0, x = xi

v^2 = k(xi^2)/m - k(x^2)/m

v = sqrt(k(xi^2)/m - k(x^2)/m)

t = sqrt(m/k)∫dx/sqrt(xi^2 - x^2)

t = arcsin(x/xi)sqrt(m/k). Therefore, if we allow ∆x = 0

t = arcsin(0)sqrt(m/k). Hence, if we want the time for one period:

t = 2pisqrt(m/k)

Is this correct? Thanks!

#### rock.freak667

Homework Helper
Yes that is correct.
But if you didn't want to integrate you could have just put F=ma=-kx
such that a= -(k/m)x
which is in the form a=-(w^2)x meaning that w=sqrt(k/m)

and the period T=2pi/w

#### jgens

Gold Member
Thanks! That's a much simpler derivation.

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