# Equation of motion of a mass-spring system

by mech-eng
Tags: equation, massspring, motion
 P: 151 hi, all. I am trying to derive the equation of motion of a mass spring system without using the energy method but I am wrong somewhere and I cant find it, can you help me find where I am wrong. Equation of motion of a simple mass spring system is indeed mx''+kx=0 but here I am thinking that when we pull the mass, motion arises from the spring force which is trying to bring back the mass and it is -kx due to our choice of negative direction but when the force is negative, i.e -kx, the acceleration x'' must also be negative because they are in the same direction and sense. Here their sense both are negative. So equation should be -mx''=-kx(sum of forces equal mass product acceleration) and thus -mx''+kx=0 Can you explain me where I am wrong?
PF Gold
P: 1,325
 Quote by mech-eng i.e -kx, the acceleration x'' must also be negative because they are in the same direction and sense.
Exactly. Therefore, writing
$$\ddot{x} = \frac{-k x}{m}$$
ensures that the acceleration ##\ddot{x}## is negative when the displacement ##x## is postive. If you add a minus sign in front of ##m \ddot{x}##, you get a positive acceleration for a positive displacement.
 Sci Advisor PF Gold P: 1,325 I should also add that the base formula is ##F=ma##. Once you have figured out what ##F## is, the equation must be applied directly, without modifying the ##ma## part.
P: 151
Equation of motion of a mass-spring system

 Quote by DrClaude Exactly. Therefore, writing $$\ddot{x} = \frac{-k x}{m}$$ ensures that the acceleration ##\ddot{x}## is negative when the displacement ##x## is postive. If you add a minus sign in front of ##m \ddot{x}##, you get a positive acceleration for a positive displacement.
It is very clear, thanks a lot.

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