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 postive. 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 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.

 

[PJC01]

Your reasoning is correct, but there is a small mistake in your equation. The correct equation of motion for a mass-spring system is indeed mx'' + kx = 0, where x'' is the acceleration and k is the spring constant. However, in your equation, you have a negative sign in front of mx'', which is incorrect.

To understand why this is incorrect, let's look at the forces acting on the mass in the system. When we pull the mass, we are applying a force in the positive direction, which we can denote as F. At the same time, the spring is exerting a force in the negative direction, which we can denote as -kx. According to Newton's second law, the sum of these forces should equal the mass times the acceleration, so we have F - kx = mx''. Notice that both F and kx are positive, so there is no need for a negative sign in front of mx''.

In summary, the equation of motion for a mass-spring system is mx'' + kx = 0, where x'' is the acceleration and k is the spring constant. The negative sign in front of kx comes from our choice of direction (positive direction is towards the pulled position, while negative direction is towards the equilibrium position), not from the force itself. I hope this helps to clarify your understanding.