SHM/Oscillation-Vertical Spring w/mass

[lilmoomoobean]

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Im a little confused about the net forces acting on a vertical spring with a mass attached to it at different times. Downward displacement is positive direction
1.@ new equilibrium pt
2.@Lowest point (most stretched)+A
3.@ Highest point (most compressed) -A

I came to the following conclusion, but I ran into a little problem...can someone tell me what I am doing wrong?

y0=equi pt of spring w/o mass, y'=new equilibrium pt with mass

1.@ new equilibrium pt...Fnet=0=Fs-mg
Fs=-ky' and Fg=mg
Since Fs-mg=0
=-ky'-mg
-ky=mg
-y=mg/k or y=mg/-k


2. @+A, lowest pt, Fs is up and (-)
Fnet=Fs-mg
Fnet=-k(y'+A)-mg
Fnet=-ky'-kA-mg
Fnet=-k(mg/-k)-kA-mg
Fnet=mg-KA-mg
Fnet=-kA ok...this makes sense because Fnet is up and that is the oppsite direction of gravity, therefore negative👍


3.@-A, highestt pt, Fs is down and (+)
Fnet=Fs+mg
Fnet=-k(y'-A)+mg
Fnet=-ky'+kA+mg
Fnet=-k(mg/-k)+kA+mg
Fnet=mg+kA+mg
Fnet=+kA+2mg??????!?!?! this does not make sense...what did I do wrong here??!?!👎

 

[Rabolisk]

Ok, so the positive direction is positive. Mg should be positive, and Fnet = mg + Fs at all points. The sign and magnitude of Fs determine the sign of Fnet. You have the sign of mg changing, which isn't right. The part that's wrong is y=mg/-k. Remember that k is always positive. You're implying that the spring would go upwards as a result of an object hanging from it. That is the source of your error in 3. In 2, you made a double error which canceled each other out. Hope that made sense!

 

[lilmoomoobean]

I get what you mean by changing the sign of mg....but I guess what I was trying to get at is that at the top, the Fs and mg are pointed in the same direction (both are pointing downwards), therefore shouldnt they be adding together (Fnet=Fs+Fg)? Also, at the bottom Fs and Fg are pointing different directions (Fg down, Fs up), making the Fnet=Fs-mg.


Can you also point out what mistake I made in #2? Or maybe if you can just re-write #2 and #3 to show me what I did wrong?

I hope you get what i am talking about....I was basically trying to draw a free body diagram at the highest point and lowest point of all of the forces that are acting on the spring plus their direction. When I try to represent what I am saying algebraically (sp?), the negative sign just start to get confusing and the more I think about it, the more confusing it gets for me.


Thanks so much

 

[Rabolisk]

Pick one direction as positive. Let's say downward is positive. Upward is negative. There are two forces, Fs and weight. Fs is -ky by definition, and the sign is important. The variable y is the displacement from the original, natural equilibrium position. Weight is mg.

1. Fnet = Fs + mg = 0
Fnet = -ky + mg = 0
y = mg/k

y is positive because m, g, and k are positive. This makes sense because the object brings the spring downward by some distance before the upward, restoring force and gravity balance out.

2. Fnet = Fs + mg = -ky + mg
Fnet = -k(mg/k + A) + mg
Fnet = -mg - kA + mg = -kA

Fnet is negative and upwards, which makes sense at the bottom amplitude. A is the displacement from the new equilibrium position, thus -kA follows Hooke's Law.

3. Fnet = Fs + mg = -ky + mg
Fnet = -k(mg/k - A) + mg
Fnet = -mg +kA + mg
Fnet = kA

It makes sense that the net force is acting downwards at the top amplitude.

You are right in that Fs and mg point to opposite directions in #2. However, that is accounted for when you substitute Fs with -ky. The sign of Fs changes as the sign of y changes. y is positive when it is below the equilibrium point, and negative in the opposite case.