Equation of motion of a mass-spring system

[mech-eng]

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 can't 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?

 

[DrClaude]

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 positive. If you add a minus sign in front of $m \ddot{x}$, you get a positive acceleration for a positive displacement.

 

[DrClaude]

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.

 

[mech-eng]

Exactly. Therefore, writing
$$
\ddot{x} = \frac{-k x}{m}
$$
ensures that the acceleration $\ddot{x}$ is negative when the displacement $x$ is positive. 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.