8-36 할리데이 11판 솔루션 일반물리학

$$ \put \begin{cases} 0 : \text{Start}\\ 1 : \text{2cm move}\\ 2 : \text{Spring Neutral Point}\\ \end{cases} $$ $$ \begin{cases} m&=20\ut{kg}\\ k&=6.0\ut{kN/m}=6.0\times10^3\ut{N/m}\\ x_0&=10\ut{cm}=0.1\ut{m}\\ f&=80\ut{N}\\ \Delta x_{0\rarr 1}&=2\ut{cm}=2\times10^{-2}\ut{m} \end{cases} $$ $$ d=-\Delta x $$ $$ \put \begin{cases} \KE : \text{Kinetic Energy}\\ \LE : \text{Elastic Potential Energy}\\ \TE : \text{Thermal Energy}\\ \end{cases} $$ $$\Sigma \Delta E=0,$$ $$\Delta \KE+\Delta \LE+\Delta \TE=0$$ $$ \begin{aligned} \KE_f&=-\Delta LE-\Delta TE\\ &~~~~~(\because v_0=0)\\ &=-\Delta \(\frac{1}{2}kx^2\)-f d\\ &=-\frac{1}{2}k\Delta \(x^2\)+f \Delta x\\ &=\frac{1}{2}k\({x_0}^2-{x}^2\)+f (x-x_0)\\ \end{aligned} $$ $$\therefore \KE(x\ut{m})=22 + 80 x - 3000 x^2\ut{J}$$ $$\ab{a}$$ $$ \begin{cases} x_a&=10\ut{cm}-2\ut{cm}\\ &=8\ut{cm}=0.08\ut{m} \end{cases} $$ $$ \begin{aligned} \KE(0.08)&=22 + 80 (0.08) - 3000 (0.08)^2\\ &=\frac{46}{5}\ut{J}\\ &=9.2\ut{J} \end{aligned} $$ $$\ab{b}$$ $$ \begin{cases} x_b&=0 \end{cases} $$ $$ \begin{aligned} \KE(0)&=22 + 80 (0) - 3000 (0)^2\\ &=22\ut{J}\\ \end{aligned} $$ $$\ab{c}$$ $$ \begin{aligned} \dyx{\KE}&=\dx(22 + 80 x - 3000 x^2)\\ &=80-6000x=0\\ \end{aligned} $$ $$x_{\max}=\frac{1}{75}\ut{m}$$ $$ \begin{aligned} \KE(x_{\max})&=22 + 80\cdot \(\frac{1}{75}\) - 3000\cdot \(\frac{1}{75}\)^2\\ &=\frac{338}{15}\ut{J}\\ &\approx 22.533333333333335\ut{J}\\ &\approx 23\ut{J}\\ \end{aligned} $$