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I've a rather simple confusion about restoring forces, and it has been bothering me for a small while now.

One of the simplest restoring forces, is the one described by Hooke's law :

$$\vec{F}=-k\vec{x}$$

Suppose, I'm stretching a spring to the right of the equilibrium, and I've defined the right side to be the positive direction, denoted by the unit vector $\hat{i}$. Hence I can write the above equation as :

$$\vec{F}=-kx\hat{i}=kx(-\hat{i})$$ This shows that the force acts to the left side, as expected. However, this is where my confusion begins.

In order to solve this equation, we compare this to Newton's second law. Hence, we write $$\vec{F}=m\ddot{x}\hat{i}=kx(-\hat{i})$$

Hence, $$m\ddot{x}=-kx$$

My problem is, why do we write $\vec{F}=m\ddot{x}\hat{i}$ and not $m\ddot{x}(-\hat{i})$ since we already know that the force works in the negative $\hat{i}$ direction ? Why do we inherently consider the force to work in the positive $\hat{i}$ direction and then write something like, the force in the positive $\hat{i}$ direction is $-kx\hat{i}$ ? This seems like saying, the force in the positive direction is negative, and so, the force must be in the negative direction.

Couldn't we have directly said that the force in the negative direction is positive, and defined $\vec{F}=-m\ddot{x}{\hat{i}}$ ?

Is this some convention in Newton's law, where $\vec{F}$ is always defined as $+ m\ddot{x}\hat{i}$, that force in the direction of displacement is assumed to be positive, and only if $m\ddot{x}\hat{i}$ becomes negative, only then we can say the force does the work in the negative direction ?

It doesn't seem to matter whether the force is acting along the left or the right, in the equation of motion, we always take $\vec{F}=m\ddot{\vec{x}}$ along whichever direction has been considered to be positive.

In my SHM sum, if I had considered the exact same system, but considered the left direction to be positive i.e $\hat{i}$, then the displacement would be negative, if I stretch the spring to the right side. However, in that case too,

$$\vec{F}=-k\vec{x}=-k(-x)\hat{i}=kx\hat{i}$$

However, what would be the LHS in this scenario. Should I write, $\vec{F}=m\ddot{x}\hat{i}$ as before ? But this would not give me the correct equation of motion, as there is a missing negative sign.

My guess is, since $x$ is negative here, the acceleration $\ddot{x}$ is also negative, and this is where the extra negative sign comes from.

Another alternative guess is that, if the force is defined to be $\vec{F}=m\ddot{x}\hat{x}$ where $\hat{x}$ is the direction of positive displacement, then in our sum, $\hat{x}=-\hat{i}$.

So, my confusion is regarding why $\vec{F}$ is taken to be positive and written as $m\ddot{x}\hat{x}$, even when we know that the force is acting in the negative direction ?

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2 Answers 2

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Ignoring the absolute directions for a moment, things accelerate in the same direction they’re pushed, and springs resist deforming. This explains the lack and presence of a minus sign, respectively, in $\vec{F}=m\ddot{\vec{x}}$ and $\vec{F}=-k\vec{x}$. That’s all you need.

The problem arises because you decide on a direction when you replace $\vec{x}$ with $x\hat{i}$. It looks like this ends up confusing you because $\hat{i}$ always points in the positive direction, whereas you know that $\vec{F}$ doesn’t; thus, it’s compelling to add or remove minus signs. But any negatives are taken care of in the sign of $x$ and $\ddot{x}$. So you must resist this compulsion to avoid a sign error.

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  • $\begingroup$ Thank you so much, but I'm still unable to resolve the confusion with the second problem. Suppose the displacement is to the right, but I take the left side to be the positive direction. In that case $\vec{x}$ is negative, and we can write it as $-x\hat{i}$. So, $\vec{F}=kx\hat{i}$. This shows that force must act to the left as expected. But now if I plug in $\vec{F}=m\ddot{x}\hat{i}$, I get the wrong equation of motion. $\endgroup$
    – RayPalmer
    Jan 3, 2022 at 16:17
  • $\begingroup$ If you also add the rule that $x$ must be a positive number, as you’ve done here, then you must manually write the equations differently piecewise for the different segments of the motion, positive and negative. I don’t recommend this “overmanaging” approach because of the confusion it can cause. $\endgroup$ Jan 3, 2022 at 16:23
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$\vec{F}$ isn't "taken to be positive", rather, the direction of force is always taken to be parallel to the direction of the acceleration. If we abided by your equation $\vec{F}=m\frac{d^2x}{dt^2}(-\hat{i})$ then the direction of the force would have the opposite sign of the acceleration, so the acceleration and the displacement would be in the same direction, which is not the form of simple harmonic motion.

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