Find Spring Constant: Prove 1/K_eff = 1/k_1 + 1/k_2

[Fanta]

Finding the given spring constant

Homework Statement

Consider the system represented on the figure, constituted by the mass m and two springs of constant k1 and k2.
(The image is attached)
Prove that:

$$\frac{1}{K_{eff}} = \frac{1}{k_{1}} + \frac{1}{k_{2}}$$

Homework Equations

$$F = -kx$$

The Attempt at a Solution

I don't know where to begin. I have to consider two different displacements: One for the first spring, and one for the second.
I think, but I am not sure, that I can consider both forces are equal.
So:

$$F1 = -k_{1} x_{1}$$

$$F2 = -k_{2} x_{2}$$

and

1)

$$F1 + F2 = 0$$

and a resultant force =

$$F_{r} = -k_{eff} x$$

2)

$$x = x1 + x2$$

I tried making a system with equations 2 and 1, but I am getting nowhere. Can anyone help?

 

[rock.freak667]

The force on both springs are equal i.e. F₁=F₂

Now the force on one spring is also equal to k_eff(x₁+x₂).

I think you can now find k_eff

 

[Fanta]

Now the force on one spring is also equal to k_eff(x₁+x₂).

can you explain why is that so, please?

 

[rock.freak667]

can you explain why is that so, please?

The force on the spring should be the same throughout.

 

[Fanta]

Like this i found it easy to do, thanks.

but isn't that the force for both springs combined?
I mean, you have the total displacement, and the Keff.

Or I can choose any of the springs, say F1 = Keff(x1+x2) ?

 

[rock.freak667]

Like this i found it easy to do, thanks.

but isn't that the force for both springs combined?
I mean, you have the total displacement, and the Keff.

Or I can choose any of the springs, say F1 = Keff(x1+x2) ?

You can choose any spring and it should work out.