8-75 할리데이 11판 솔루션 일반물리학

$$ \begin{cases} R&=18.0\ut{m}\\ \theta_i&=0\\ y_i&=R\\ v_i&=0\\ \end{cases} $$ $$ \begin{aligned} \Delta y &= -(R-R\cos\theta_f)\\ \end{aligned} $$ $$\Sigma \Delta E=0,$$ $$\Delta \KE+\Delta \GE=0$$ $$ \begin{aligned} \Delta \KE&=-\Delta \GE\\ \Delta \(\frac{1}{2}m{v}^2\)&=-\Delta (mgy)\\ \Delta \({v}^2\)&=2g(-\Delta y)\\ \end{aligned} $$ $$ \begin{aligned} {v_f}^2&={v_i}^2+2gh\\ &=2g(-\Delta y)\taag1\\ \end{aligned} $$ $$ \begin{aligned} \Sigma F_R&=mg\cos\theta-N=\frac{mv^2}{R},\\ N_f&=mg\cos\theta_f-\frac{m{v_f}^2}{R}=0\\ \end{aligned} $$ $$ \begin{aligned} mg\cos\theta_f&=\frac{m{v_f}^2}{R}\\ gR\cos\theta_f&={v_f}^2\\ gR\cos\theta_f&=2g(-\Delta y)\\ R\cos\theta_f&=2(R-R\cos\theta_f)\\ \cos\theta_f&=2-2\cos\theta_f\\ \cos\theta_f&=\frac{2}{3}=\frac{y}{R} \end{aligned} $$ $$ \begin{aligned} y&=\frac{2}{3}R\\ &=12\ut{m}\\ &=12.0\ut{m}\\ \end{aligned} $$