text: ## 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
metadata: {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}