10-44 할리데이 11판 솔루션 일반물리학

$$ \begin{cases} d&=0.50\ut{m}\\ m&=0.30\ut{kg}\\ M&=\Sigma m=4m \end{cases} $$ $$\ab{a}$$ $$I=\Sigma mr^2,$$ $$ \begin{cases} r_1&=0\\ r_2&=d\\ r_3&=d\\ r_4&=\sqrt2 d\\ \end{cases} $$ $$ \begin{cases} I_1&=0\\ I_2&=md^2\\ I_3&=md^2\\ I_4&=m(\sqrt2 d)^2=2md^2\\ \end{cases} $$ $$ \begin{aligned} \Sigma I&=I_1+I_2+I_3+I_4\\ &=4md^2\\ &=0.3\ut{kg\cdot m^2} \end{aligned} $$ $$\ab{b}$$ $$ \put \begin{cases} \RE : \text{Rotational Kinetic Energy}\\ \KE : \text{Translational Kinetic Energy}\\ \GE : \text{Gravitational Potential Energy}\\ \end{cases} $$ $$ \begin{cases} y_{\com i}&=\frac{d}{2}\\ y_{\com f}&=-\frac{d}{2}\\ \end{cases} $$ $$\Delta y_{\com}=-d\taag1$$ $$ \begin{aligned} r_{\com}&=\sqrt2d, \end{aligned} $$ $$ \begin{aligned} v^2&=(r\omega)^2\\ &=2d^2\omega^2\taag2 \end{aligned} $$ $$\Delta \Sigma E=0,$$ $$ \begin{aligned} 0&=\Delta\KE+\Delta\RE-\Delta \GE\\ &=\KE_f+\RE_f-\Delta \GE\\ &=Mv^2+I\omega^2-2Mg\Delta y_{\com}\\ &=(4m)(2d^2\omega^2)+(4md^2)\omega^2-2(4m)g(-d)\\ &=3d\omega^2+2g\\ \end{aligned} $$ $$ \begin{aligned} \omega&=\sqrt\frac{2g}{3d}\\ &=2\sqrt\frac{g}{3}\\ &\approx 3.6160106931995286\ut{rad/s^2}\\ &\approx 3.6\ut{rad/s^2}\\ \end{aligned} $$