8-77 할리데이 11판 솔루션 일반물리학

$$ \begin{cases} m&=5.0\ut{kg}\\ k&=425\ut{N/m}\\ x&=6.0\ut{cm}=0.06\ut{m}\\ \mu&=0.25\\ g&=9.80665\ut{m/s^2} \end{cases} $$ $$ \put \begin{cases} \KE : \text{Kinetic Energy}\\ \LE : \text{Elastic Potential Energy}\\ \TE : \text{Thermal Energy}\\ \end{cases} $$ $$\ab{a}$$ $$ \begin{aligned} W_s&=-\Delta \LE\\ &=\LE_i-\LE_f\\ &=-\LE_f\\ &=-\frac{1}{2}kx^2\\ &=-\frac{153}{200}\ut{J}\\ &=-0.765\ut{J}\\ &\approx -0.77\ut{J} \end{aligned} $$ $$\ab{b}$$ $$ \begin{aligned} \Delta \TE&=fS=f\Delta x\\ &=\mu m g\Delta x\\ &=\frac{3g}{40}\\ &=0.73549875\ut{J}\\ &\approx 0.74\ut{J}\\ \end{aligned} $$ $$\ab{c}$$ $$\Sigma \Delta E=0,$$ $$\Delta \KE+\Delta \TE+\Delta \LE=0$$ $$ \begin{aligned} \Delta \TE+\Delta \LE&=-\Delta \KE\\ \mu m g\Delta x+\frac{1}{2}k\Delta(x^2)&=-\Delta\(\frac{1}{2}m{v}^2\)\\ \mu m g x_f+\frac{1}{2}k{x_f}^2&=-\frac{1}{2}m\Delta\({v}^2\)\\ &=\frac{1}{2}m\({v_i}^2-{v_f}^2\)\\ &=\frac{1}{2}m{v_i}^2\\ \end{aligned} $$ $$ \begin{aligned} v_i&=\sqrt{\frac{x(2\mu mg+k x)}{m}}\\ &=\frac{1}{10} \sqrt{3 g+\frac{153}{5}}\\ &\approx 0.7747254352349612\ut{m/s}\\ &\approx 0.77\ut{m/s}\\ &\approx 77\ut{cm/s}\\ \end{aligned} $$