Coordinate System-Spring Vertical

[samtouchdown]

Coordinate System--Spring Vertical

Hi! This is a question on the use of a coordinate system.
In the princeton review, I don't understand the coordinate system they are using it; it doesn't make sense. That is, for a vertical spring, the net force on the mass is kx-mg. But, shouldn't it be mg-kx? The component of spring vertical is -kx, not +kx. What type of coordinate system would have +kx? This may be a trivial matter, but any help to ease my slight confusion would help! Thanks.

 

[haruspex]

If sense is used consistently, the displacement should be measured in the same direction as the force. You don't supply a diagram. The only way I can imagine the displacement and force would be positive at the same time is if the spring is considered part of the mass, and the displacement is that of the other end of the spring. That would be unusual.
Alternatively, and again unusually, displacement is being measured in one sense and force in the opposite one.

 

[Chirag B]

Haruspex is correct in his response. The idea is that the force given by Hooke's Law for Springs counters the motion. This is the reason for the negative sign. So here, the object is moving downward. Let us choose the downward direction for $+\hat{i}$. Then, $mg$ is "positive", and the force that counters it is the one given by Hooke's Law for Springs, $-kx$. So the net force can be found using Newton's Second Law: $F_{net} = F_{g} + F_{s} = mg+(-kx)$ which can be simplified down to $mg-kx$, as you asked for. Just realize that it counters the motion. It moves downward, so Hooke's Law tells you that $F_{spring}$ counteracts this, and thus carries the negative sign. By convention, Princeton Review chooses down $(\downarrow)$ to be the direction for $+\hat{i}$. They mention this somewhere in the early stages of the book. Hope this helps!

 

[samtouchdown]

But they do kx-mg in the princeton review.

 

[Chirag B]

Oh I see what you're saying. So they take up to be positive then, that's all.

 

[samtouchdown]

Chirag B: I don't understand how they take it to be upward. Fspring=-kx always.
In this case, they define up to be positive and down to be negative. Suppose, the spring is stretched 2 m, then the displacement vector is obviously=-2i. So, then this "kx", which they use, would yield -2k, still in het direction of the stretch, which is obviously wrong.

 

[Chirag B]

I'm not sure I understand what you're saying Princeton Review is doing. Can you possibly scan a picture so others understand how this works?

 

[samtouchdown]

Here. Please tell me if the image
is of bad quality.

 

[haruspex]

Here. Please tell me if the image
is of bad quality.

I can't read it. Needs about nine times the pixels.