Determining equilibrium position between two springs

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Homework Statement

see attachment ***indicates correct answer

Homework Equations

F=ks

The Attempt at a Solution

I do not understand how this works, and I haven't been able to find any examples of this.

 

[bigguccisosa]

Equilibrium would imply that the net force on the mass is zero. The only 2 forces are the forces from either spring. So you must find the position at which the spring forces cancel out. Remember that force from a spring is F = -kx, where x is displacement from equilibrium position and is a vector quantity.

 

[RedDelicious]

As the above poster said, the net force on the mass is going to be zero because it's in equilibrium, meaning you can equate the two spring forces. That will give you one equation with two unknowns, meaning you're missing some piece of information. When it's not obvious, it's usually some constraint equation you're overlooking, and in this case it will be related to the total length of both springs. Because the forces are in equilibrium and you're given their lengths when not stretched, you can deduce a piece of information about their lengths in the equilibrium state that will allow you to solve the equation and find the block's position.

 

[Scientifically]

distance equals spring length + spring displacement at equilibrium: d=1+x1
distance equals total system length - length of second spring and its displacement: d=3-(1+x2)
equate both expressions: 1+x1=3-1-x2 isolate x2: x2=1-x1

forces are equal at equilibrium => k1x1=k2x2 isolate x2: x2=(k1x1)/k2

equate both expressions of x2:
(k1x1)/k2 = 1-x1 => x1=k2/(k1+k2)

=> d=1+x1= 1+k2/(k1+k2) = 1+ 300/400 = 1.75 m