The figure below shows two springs connected in parallel. This combination can be thought of as being equivalent to a single spring having an effective force constant keff. Obviously, the effective force constant must be related to the force constants k1 and k2 of the individual springs. We will assume that the springs have no significant mass. Suppose we stretch the combination a total distance x from its equilibrium position.(Figure 1)



If you connect a spring of force constant 35.0N/cm in parallel with one of force constant 55.0N/cm , what is the force constant of the single spring that will be equivalent to this combination?

x₁=x₂=x

F=F₁+F₂
F/x= F₁/x +F₂/x =
F₁/x₁+F₂/x₂ =>
k=k₁+k₂= 35+55 = 90 N/m

To find the equivalent force constant of the parallel combination of two springs, we can use the formula:

1/keff = 1/k1 + 1/k2

Where keff is the effective force constant and k1, k2 are the force constants of the individual springs.

In this case, k1 = 35.0 N/cm and k2 = 55.0 N/cm.

Let's plug in the values and calculate:

1/keff = 1/35.0 + 1/55.0

To add these fractions, we need a common denominator, which is the product of the denominators:

1/keff = (55/55)*(1/35) + (35/35)*(1/55)
= 55/1925 + 35/1925
= 90/1925

Now, we can invert both sides of the equation to solve for keff:

keff = 1925/90

Simplifying the fraction:

keff = 385/18

Therefore, the force constant of the single spring that is equivalent to the parallel combination of the springs is 385/18 N/cm.