Why Does the Restoring Force Use k1x + k2x in Spring Oscillations?

[jsmith613]

Looking at the three diagrams we can see that there are three possible situations

(a) mass is to the right of eqm
(b) mass is at eqm
(c) mass is to the left of eqm
(eqm = equilibrium)

Lets look at position (a)

If we consider the tension in both springs:
the tension in the spring on the left has increased by k₁x
the tension in the spring on the right has decreased by k₂x

So the overall restoring force SHOULD BE k₁x - k₂x
(the spring on the left is trying to pull it back to the left and the spring on the right is trying to pull it the right hence the forces act in opposite directions)

BUT according to my book the overall restoring force is k₁x + k₂x...how?

Any help is greatly appreciated

exact quote:

Book
Consider the mass at some point during motion. Let its displacement from eqm be x at that point. One of the two springs has been extended by x and the other has been shortened by x. So compared with eqm one spring has extra tension k₁x and the other string has its tension reduced by k₂x.
The spring constants for the individual springs are k₁ and k₂.
The extra tension from one spring combines with the reduced tension from the other to give a restoring force of k₁x + k₂x.
The restoring force can be written as: F = -kx
where k = k₁ + k₂
The - sign indicates it acts towards eqm

 

[Integral]

The compressed spring is not pulling away from the eqm but pushing towards. IE it is under compression.

 

[jsmith613]

The compressed spring is not pulling away from the eqm but pushing towards. IE it is under compression.

oh right
thanks

 

[jsmith613]

The compressed spring is not pulling away from the eqm but pushing towards. IE it is under compression.

why is tension not present here?
(I don't understand the explanation given)?

 

[Doc Al]

why is tension not present here?
(I don't understand the explanation given)?

I don't understand the choices. There are two forces acting on the hanging mass: gravity and the force from the spring. Of course, a stretched spring exerts a tension. So I don't know the intention of having 'tension' vs 'spring' as separate choices: you could call that force the spring force or the tension force exerted by the spring.

And the given explanation reads like gibberish. Where is this question from?

 

[willem2]

The compressed spring is not pulling away from the eqm but pushing towards. IE it is under compression.

Even if both springs are under tension all of the time, the restoring force would still be $k_1x + k_2x$. There's an increased force to the left, because the left spring pulls harder and a decreased force to the right because the right sprint pulls less hard than in the equilibrium position. This will give the same effect as an increased force to the right.

 

[FeX32]

A linear spring pulled from equilibrium exerts a force k*x. A linear spring compressed from equilibrium exerts a force k*x. Thus, the force is k*x+k*x.

I'm not sure where the issue lays

 

[FeX32]

By the way, that attached .png is garbage, no offence.