8-45 할리데이 11판 솔루션 일반물리학

$$ \put \begin{cases} \GE : \text{Gravitational Potential Energy}\\ \LE : \text{Elastic Potential Energy}\\ \end{cases} $$ $$ \begin{cases} x_i&=0\\ kx_f&=mg\\ \Delta x&=\abs{\Delta h}=-\Delta h \end{cases} $$ $$ \begin{aligned} -\Delta \GE &= -\Delta (mgh)\\ &=-mg\Delta h\\ &=mg(-\Delta h)\\ &=mg\Delta x\\ &=mg(x_f-x_i)\\ &=mgx_f\\ &=mg\(\frac{mg}{k}\)\\ &=\frac{(mg)^2}{k}\\ \end{aligned} $$ $$ \begin{aligned} \Delta \LE&=\Delta \(\frac{1}{2}kx^2\)\\ &=\frac{1}{2}k\Delta \(x^2\)\\ &=\frac{1}{2}k\({x_f}^2-{x_i}^2\)\\ &=\frac{1}{2}k{x_f}^2\\ &=\frac{1}{2}k\(\frac{mg}{k}\)^2\\ &=\frac{(mg)^2}{2k}\\ \end{aligned} $$ $$\therefore -\Delta \GE : \Delta \LE = 2 : 1$$ $$ \begin{aligned} \text{Work by } F_N \end{aligned} $$