# Why must the spring forces of springs in series be equal?

I was learning about two springs in series connected to a wall on one side, and a block of mass m on the other.

Model:

WALL -- Spring 1 -- Spring 2 -- mass

They say that the force on the mass, $$F_m = -k_2x_2 = F_2$$, and the force on spring 2 is $$F_1 = -k_1x_1$$.

They also say that $$F_1 = F_2 = F_m$$. But what I don't understand is that how come Newton's third law isn't being accounted for. Wouldn't the spring force of the spring on the mass also correspond to an equal but opposite force on the spring of $$k_2x_2$$, and the total force on spring 2: $$k_2x_2 - k_1x_1 = 0$$, so there is no net force on the spring?!

Overall, I don't know where they got the result or how they got the result that $$F_1 = F_2 = F_m$$

• Could it be that the problem is assuming that the springs are massless - and therefore must have a net force of 0?
– user364438
Jun 8, 2023 at 16:20
• Yes: your comment is correct. It is common to use the massless spring approximation, which is reasonably good in a lot of situations. Jun 8, 2023 at 16:22

Springs in series in equilibrium experience the same tension (force) throughout while the displacements (the amount each stretches) depends upon the individual spring constants. This is necessary in order to satisfy both Newton’s 2nd and 3rd laws.

See the free body diagrams below showing the Newton 3rd law pairs and that the net force acting on the spring-mass system by the wall and external force on the mass system is zero satisfying Newton’s 2nd law for equilibrium.

Hope this helps.

• Bob D, that's a good explanation and a really nice diagram, deserving of the upvote that I gave it. Jun 8, 2023 at 18:36
• @DavidWhite thanks for that Jun 8, 2023 at 18:52
• Wait, doesn't this assume that the springs are massless? And what if the springs are not in equilibrium but moving?
– user364438
Jun 8, 2023 at 19:12
• @EmilSriram yes it assumes the springs are massless since you provided no info on mass. And yes equilibrium is assumed since you said the system is attached to a wall and said nothing about harmonic motion. If you wished to include those things post a different question Jun 8, 2023 at 19:19
• Alright. Seems a bit passive aggressive, but okay. But seriously, greatly appreciate your help!
– user364438
Jun 8, 2023 at 19:47

The mass exerts a force on spring 2, the modulus of which is equal to $$$$F_m=k_2x_2$$$$ The spring 2, on the other hand, exerts a force on the spring 1 uqual to $$$$F_{2\rightarrow 1}=k_1x_1$$$$ Meanwhile, the spring 1 exerts a force on the spring 2: $$$$F_{1\rightarrow 2}=k_2x_2$$$$ However, for the third principle these two forces must be equal. So you get the result.

• this makes no sense. why does the third statement work?
– user364438
Jun 8, 2023 at 16:34
• The third law of dynamics is a first principle. We know from experience that, if a body (like spring 1) applies a force on a second body (like spring 2), then the second body applies a force equal and opposite to the first body, without exceptions. I hope this answer your doubt Jun 8, 2023 at 16:40
• No but why does the spring 1 apply k2x2 force on spring 1. I thought that the spring 2 applies that to the mass?
– user364438
Jun 8, 2023 at 16:50
• Ah I think I understood now. According to your picture, the spring 2 applies a force from right to left to the mass, and also a force from left to right to the spring 1; this last force is k1x1. The spring 1, instead, applies only a force from right to left to the spring 2 Jun 8, 2023 at 17:08