How Do You Calculate Spring Extension in a Series Configuration?

[bmxicle]

Homework Statement

4 springs with a mass on each end are connected in series as below:

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m1
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m2
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m3
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m4

All the masses are mass m, the length of each spring is 1, and the spring constant is k, find the extension of each spring.

Homework Equations

f = ky

The Attempt at a Solution

So basically if i label each spring extension as $$y_1, \ y_2, \ y_3, \ y_4$$ then the only forces are the spring pulling from above, the spring pulling from below and the force of gravity on each mass. So for example for the second mass m2 from the top the equilibrium equation would be:

$$0 = ky_2 - ky_3 -mg$$

If this is right I can solve the rest of the problem easily, but I'm just getting thrown off by my use of Newton's second law here.

 

[Villyer]

0=ky2−ky3−mg

This seems to be an accurate statement.

Now how can you rewrite the ky2 and ky3s into the variables you do know, which are m, k and g?

 

[Redbelly98]

... the only forces are the spring pulling from above, the spring pulling from below and the force of gravity on each mass. So for example for the second mass m2 from the top the equilibrium equation would be:

$$0 = ky_2 - ky_3 -mg$$

Yes, you are on the right track.

 

[bmxicle]

Thanks, I can solve the rest of it pretty easily now. I think I was just over thinking which springs were pulling where.

 

[asteriatic]

I would say that your approach is correct in using Newton's second law to solve this problem. You have correctly identified the forces acting on each mass and set up the equilibrium equation for each one. This is a good application of Hooke's law, which states that the force exerted by a spring is directly proportional to the amount it is stretched or compressed. In this case, the force exerted by each spring is equal to its spring constant times its extension. By setting up these equilibrium equations, you can solve for the extensions of each spring and verify that they are in line with what is expected according to Hooke's law. This is a good sanity check to make sure that your approach and calculations are correct.