Physics Assessment #38: Spring constant

[LuminousTruth]

What is the effective spring constant of the system of two springs shown in Fig 1?

A) K
B) 2K
C) K/2
D) 0

The correct answer was B



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I thought that when the mass moved a displacement "X", the two spring constants will act in opposite directions, one spring constant lessening the effect of the second, making me choose C. Can someone explain to me why that reasoning is incorrect?

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[Ace-Co-A]

For the two sides of the block,
F1 = -kx1
F2 = -kx2

If I were to pull the block some displacement x in either direction, then the displacement with respect to the other side of the block must be -x. The sum of the forces exerted on the block by the springs must equal the net force exerted on the block. One of the forces must be negative because it is directed opposite that of the force of the compressed spring (or vice versa).

F1 - F2 = Fnet
-kx - [-k(-x)] = -knet*x

Solving, we get: -kx - kx = -knet*x
The displacement "x" can be cancelled because it is the same: -k - k = -knet
The negative sign can be cancelled: k+k = knet
So, therefore, the effective spring constant knet = 2k

 

[MangoPlant]

Simple way: Imagine moving the block to the right. The spring on the right will push to the left and the spring on the left will pull to the left. Notice that they are both exerting forces in the same direction, thus they work together must be stronger (have a higher K) as a system than individually. The only answer choice that is greater than K is 2K so that is your answer.