Answer
(a) The spring must be compressed a distance of 6.32 cm.
(b) The spring will be compressed a maximum distance of 11.6 cm.
Work Step by Step
(a) $\frac{1}{2}kx^2 = 3.20~J$
$x^2 = \frac{6.40~J}{k}$
$x = \sqrt{\frac{6.40~J}{1600~N/m}}$
$x = 0.0632~m = 6.32~cm$
The spring must be compressed a distance of 6.32 cm.
(b) The initial potential energy of the book will be equal to the potential energy stored in the spring when it is compressed a distance of x. Note that the initial potential energy of the book depends on the total change of height 0.800 m + x.
$\frac{1}{2}kx^2 = mg(0.800~m+x)$
$\frac{1}{2}kx^2 - mgx - mg(0.800~m) = 0$
$\frac{1}{2}(1600~N/m)~x^2 - (1.20~kg)(9.80~m/s^2)~x - (1.20~kg)(9.80~m/s^2)(0.800~m) = 0$
$(800~N/m)~x^2 - (11.76~N)~x - (9.408~J) = 0$
We can use the quadratic formula to find $x$.
$x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$
$x = \frac{(11.76)\pm \sqrt{(-11.76)^2-(4)(800)(-9.408)}}{(2)(800)}$
$x = 0.116~m, -0.101~m$
Since the negative value is unphysical, the solution is $x = 0.116~m$ which is 11.6 cm
The spring will be compressed a maximum distance of 11.6 cm.
