7-40 할리데이 11판 솔루션 일반물리학

$$ \begin{cases} m&=3.20\ut{kg}\\ k&=650\ut{N/m}\\ x_i&=0.400\ut{m} \end{cases} $$ $$\ab{a}$$ $$K=\frac{1}{2}mv^2,W_s=\frac{1}{2}kx^2,$$ $$ \begin{aligned} K&=W_s\\ \frac{1}{2}mv^2&=\frac{1}{2}kx^2\\ \end{aligned} $$ $$ \begin{aligned} v&=x\sqrt{\frac{k}{m}}\\ &=\sqrt{\frac{65}{2}}\ut{m/s}\\ &\approx 5.70087712549569\ut{m/s}\\ &\approx 5.70\ut{m/s}\\ \end{aligned} $$ $$\ab{b}$$ $$ \begin{aligned} W_s &= \frac{1}{2}kx^2\\ &= 52\ut{J} \end{aligned} $$ $$\ab{c}$$ $$ \begin{aligned} P&=\vec F\cdot \vec v\\ &=F_iv_i\\ &=0~~(\because v_i=0) \end{aligned} $$ $$\ab{d}$$ $$ \begin{aligned} P&=\vec F\cdot \vec v\\ &=F_0v_0\\ &=0~~(\because F_0=0) \end{aligned} $$ $$\ab{e}$$ $$\frac{1}{2}m{v_f}^2=\frac{1}{2}k{x_i}^2-\frac{1}{2}k{x_f}^2$$ $$v_f=\sqrt{\frac{k({x_i}^2-{x_f}^2)}{m}}$$ $$ \begin{aligned} P(x)&=\vec F\cdot \vec v\\ &=F_f\cdot v_f\\ &=(kx)\cdot\sqrt{\frac{k({x_i}^2-{x}^2)}{m}}\\ &=325x\sqrt{\frac{65(4-25 x^2)}{2}}\\ \end{aligned} $$ $$ \begin{aligned} \frac{\dd P}{\dd x}&=\frac{\dd }{\dd x}\(325x\sqrt{\frac{65(4-25 x^2)}{2}}\)\\ &=\frac{325 \(2-25 x^2\)}{\sqrt{\frac{2}{65}-\frac{5x^2}{26}}}=0\\ \end{aligned} $$ $$ \begin{aligned} x_{P\max}&=\frac{\sqrt2}{5}\ut{m}\\ &\approx 0.28284271247461906\ut{m}\\ &\approx 0.283\ut{m}\\ &\approx 283\ut{mm}\\ \end{aligned} $$