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## Homework Statement

For the following arrangement of two springs, determine the effective spring constant, k

_{eff}. This is the force constant of a single spring that would produce the same force on the block as the pair of springs shown for this case.

(Spring 1 is attached to Spring 2 which is attached to a block of mass m)

Solution Provided by Book:

Imagine that the block was displaced a distance x to the right of its equilibrium position. Let x

_{1}be the distance that the first spring is stretched and let x

_{2}be the distance that the second spring is stretched. Then [itex] x = x_1+ x_2 [/itex]. But [itex] x_1 = \frac {-F}{k_1} [/itex] and [itex]x_2 = \frac{-F}{k_2} [/itex], so

[itex]\frac{-F}{k_1}+ \frac{-F}{k_2} = x[/itex]

[itex]-F(\frac{1}{k_1}+ \frac{1}{k_2}) = x[/itex]

[itex]F = \frac{1}{\frac{1}{k_1} + \frac{1}{k_2}}x[/itex]

[itex]F = -\frac{k_1k_2}{k_1+k_2}x[/itex]

Therefore,

k

_{eff}= [itex]\frac{k_1k_2}{k_1+k_2}[/itex]

## Homework Equations

[/B]

F=-kx

## The Attempt at a Solution

Basically I'm trying to understand how the solution they gave works. I can follow the logic up until the last line of the solution, k

_{eff}= [itex]\frac{k_1k_2}{k_1+k_2}[/itex].

I understand that each spring exerts the same amount of Force. This is because as x, in F=-kx, increases, k compensates by decreasing.

**When I began I thought**

F

_{new}= k

_{eff}* x

I can solve for x with

[itex]x = x_1 + x_2[/itex]

[itex]x_1 = \frac{-F_1}{k_1}[/itex]

[itex]x_2 = \frac{-F_2}{k_2}[/itex]

This leads me to

[itex]\frac{-F_1}{k_1}+ \frac{-F_2}{k_2} = x[/itex]

and if you factor out the F(which you can do because the springs exert the same force) it equals

[itex]-F(\frac{1}{k_1}+ \frac{1}{k_2}) = x[/itex].

Moving it to the other side yields

[itex]F = -\frac{1}{\frac{1}{k_1} + \frac{1}{k_2}}x[/itex]

And if you add the fractions they equal

[itex]F = -\frac{k_1k_2}{k_1+k_2}x[/itex]

but then they set [itex]F = -\frac{k_1k_2}{k_1+k_2}x[/itex] equal to

**F**and come up with the final answer

_{new}= k_{eff}* xk

_{eff}= [itex]\frac{k_1k_2}{k_1+k_2}[/itex]

F

_{new}does not equal F(The force of one spring). How can they set the equations equal to each other?

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