Combinations of Springs
Springs in Series
- Consider two springs with force constants k1 and k2 connected in series
supporting a load F = mg.
- Let the force constant of the combination
be represented by k
- For the combination, supporting the load F=mg:
k1
F = kx
( where x = the total stretch )
and
x=
F
k
- For each spring
- the bottom supports mg=F and stretches by x1
F = k1 x1
k2
F
or x1 =
k1
- the top spring support mg plus the weight
of the bottom spring (which is negligible Thus F is the stretching force for both springs)
F = k2 x2 or x2 =
m
F
k2
F = mg
- The total stretch
x = x1 + x2
and
or
F F F
= +
k k1 k2
1 1 1
= +
k k1 k2
Springs in Parallel
- Consider two springs with force constants k1 and k2 connected in parallel
supporting a load F = mg.
- Let the force constant of the combination
be represented by k
- For the combination supporting the load F=mg:
k1
k2
F = kx
( where x = the total stretch )
- The two individual springs both stretch by x
but share the load (F = F1+F2) and
F1 = k1 x while
F2 = k2 x
- Thus the total force is
F = F1 + F2
or
F2
F2
m
kx = k1 x + k2 x
F = mg
and
k = k1 + k2