## 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
The spring force:
$$F=kx$$$where $$k$$ is the spring constant, and $$x$$ is the spring’s displacement. 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
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