## Spring Force

What force is required to compress a 200 mm free length spring to 110 mm if the 45 mm diameter helical compression spring has a 2.5 mm wire diameter and 10 coils? Assume the shear modulus of elasticity is 400 N/mm^2.

##
__
__**Hint**

**Hint**

The spring force:

$$$F=kx$$$

where
$$k$$
is the spring constant, and
$$x$$
is the spring’s displacement.

##
__
__**Hint 2**

**Hint 2**

$$$k=\frac{d^4G}{8D^3N}$$$

where
$$G$$
is the shear modulus of elasticity,
$$d$$
is the wire diameter,
$$D$$
is the mean spring diameter, and
$$N$$
is the number of active coils.

A spring’s deflection and force are related by
$$F=kx$$
where the spring rate (spring constant)
$$k$$
is given by:

$$$k=\frac{d^4G}{8D^3N}$$$

where
$$G$$
is the shear modulus of elasticity,
$$d$$
is the wire diameter,
$$D$$
is the mean spring diameter, and
$$N$$
is the number of active coils.

$$$k=\frac{(2.5mm)^4(400N/mm^2)}{8(45mm)^3(10)}=\frac{15,625N\cdot mm^4}{7,290,000mm^5}=0.00214\:\frac{N}{mm}$$$

Finally, to find the force required to compress the spring:

$$$F=0.00214\frac{N}{mm}\times (200-110)mm=0.19\:N$$$

0.19 N