Question

Hook's law describes an ideal spring. Many real springs are better described by the restoring force (FSp)s=−kΔs−q(Δs)^3, where q is a constant. Consider a spring with k = 250 N/m and q = 900 N/m3.
(a) How much work must you do to compress this spring 15cm? Note that, by Newton's third law, the work you do on the spring is the negative of the work done by the spring.

(b) By what percent has the cubic term increased the work over what would be needed to compress an ideal spring?

Answer 1

Answer

given,

k = 250 N/m

q = 900 N/m³

(FSp)s=−kΔs−q(Δs)^3

work done = Force x displacement

$W = \int {F. dx}$

limits are x = 0 to x = 0.15 m

work done

$W = \int_0^{0.15} (k x + q x^3)\ dx$

$W = [\dfrac{kx^2}{2}+\dfrac{qx^4}{4}+ C]_0^0.15$

$W = \dfrac{300\times 0.15^2}{2}+\dfrac{900\times 0.15^4}{4}$

W = 3.375 + 0.1139

W = 3.3488 J

b) % cubic term =$\dfrac{0.1139}{3.3488}$

   % cubic term =$3.4\ %$