Answer
a) $k=7238$ lb-in
b) $W=904.75$ in-lb and $2714$ in-lb
Work Step by Step
a) We know that Hooke's Law states that $F=k x$
Then, $k=\dfrac{F}{x}=\dfrac{21714}{3}=7238$ lb-in.
Thus, $F=7238 x$
b) i) To calculate the work done, the limits for $x$ for the first inch will be from $0$ to $0.5$ such that:
$W=\int_0^{0.5} 7238 x dx=7238[ \dfrac{x^2}{2}]_0^{0.5}$
or, $W=3619[ (0.5)^2-0] =904.75$ in-lb
ii) To calculate the work done, the limits for $x$ for the second inch will be from $0.5$ to $1$ such that:
$W=\int_{0.5}^1 7238 x dx=7238[ \dfrac{x^2}{2}]_{0.5}^1$
or, $W=3619[1- (0.5)^2] =2714$ in-lb
