Classical mechanics:effective spring const

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Homework Statement

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What is the effective spring constant for the system of the two springs, perfect pulley, and string shown on the left for it to be modeled by just one spring (constant keff) as shown on the right? Use only the variables k1 and k2 in your answer.

The Attempt at a Solution

I know that the springs in parallel get added because they undergo same change in distance and springs in series have the same effective constant as resistors in parallel .They experience equal forces.
I don't understand that will happen in the given sitch ,will the lower spring (2) stretch? will the upper spring stretch too?(because we are pulling down) how will I write the equations? I know that the force experienced by both is equal.

 

[gneill]

Both spring will stretch.

Work out the forces that each spring will experience. Draw a FBD for the pulley, identify the tensions.
Consider using the superposition principle to figure out by how much the "F end" of the lower string will move.

 

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how much the "F end" of the lower string will move.

what's F end?

 

[gneill]

what's F end?

The end where the force "F" is applied to the string.

 

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identify the tensions.

since the system is in equilibrium,
$F=k_1x_1-k_2x_2$x_1 and x_2 should be equal?
( taking down as +ve)

using the superposition principle to figure out by how much the "F end" of the lower string will move.

you mean to think as if F is applied only on lower spring?
$F=x_2k_2$
$x_2=F/k_2$

 

[gneill]

since the system is in equilibrium,
$F=k_1x_1-k_2x_2$x_1 and x_2 should be equal?
( taking down as +ve)

I don't think there's any guarantee of that. And in fact it would take a special relationship between k1 and k2 for that to be true. The pulley system here has some mechanical advantage involved.

you mean to think as if F is applied only on lower spring?
$F=x_2k_2$
$x_2=F/k_2$

Yes, that's one case, so you know how much the rope will move due to spring k2 stretching with force F. Next consider the net force that's applied to the k1 spring and how much it must stretch as a result. If the pulley descends by that much, how much does the string descend on the "F end"? Note that the pulley can turn! You might want to think of the spring k2 being fixed for this case, that is, replaced by a fixed length string.

 

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Next consider the net force that's applied to the k1 spring and how much it must stretch as a result

$x_1=(F+x_2k_2)/k_1$

If the pulley descends by that much, how much does the string descend on the "F end"

the F end will descend by $x_1+x_2$
$F=(x_1+x_2)k_e$
surely I am making a mistake because in the k_eff term the coefficient of k_2 is 4 not 2

 

[gneill]

I think you're trying to combine steps and incorporating an incorrect assumption.

First, what is the tension on both sides of the string passing over the pulley? Then using an FBD of the pulley, what must be the force pulling down on spring k1?

Next, Suppose the pulley drops by some amount Δx1, how far does the string end move? Be careful here!

 

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Suppose the pulley drops by some amount Δx1, how far does the string end move?

thinking the lower spring to be just a fixed string. if the pulley descends by x_1 , the string will slack by x_1 so to be taut the F end will descent by 2x_1.
this gives me the correct answer
but will this "slack"- compression of the k2 spring not disturb our presumption of superposition?

 

[gneill]

thinking the lower spring to be just a fixed string. if the pulley descends by x_1 , the string will slack by x_1 so to be taut the F end will descent by 2x_1.
this gives me the correct answer
but will this "slack"- compression of the k2 spring not disturb our presumption of superposition?

Nope. That's a nice thing about superposition for a linear system. You can "suppress" parts of the system and analyze what the rest is up to and how it contributes to the whole.