# S.H.M. of helical spring.

1. Apr 6, 2013

### SherlockOhms

We've just begun studying oscillatory motion and recently derived the differential equation of motion for a helical using Hooke's law and Newton's second law of motion.

The differential equation of motion for the helical spring is (dx/dt)^2 + (k/m)x = 0. In our lecture notes it says that the general solution to the above equation is x = Acos(ωt +θ), where θ is the phase constant. From this you can find the various equations for T, a and ω. Also, the notes go on to state that if θ = -pi/2 then Acos(ωt +θ) becomes Asin(ωt), which makes sense.

However, in our text book it says that x = Asin(ωt +θ) is the general solution to the differential equation of motion and that is θ pi/2 then Asin(ωt +θ) becomes Acos(ωt), this also makes sense.

So, I was just wondering is the general solution arbitrary or is there a mistake in either the notes or text?

2. Apr 6, 2013

### Staff: Mentor

Realize that cos(ωt) and sin(ωt) are the same except for a phase shift. The general solutions are equivalent.

cos(x) = sin(x + π/2)

sin(x) = cos(x - π/2)

3. Apr 6, 2013

### SherlockOhms

Cool. Thanks for that.