11-3 할리데이 11판 솔루션 일반물리학

$$ \begin{cases} m&=120\ut{g}=0.12\ut{kg}\\ I&=950\ut{g\cdot cm^2}=9.5\times10^{-5}\ut{kg\cdot m^2}\\ r&=3.2\ut{mm}=3.2\times10^{-3}\ut{m}\\ y&=120\ut{cm}=1.2\ut{m}\\ v_0&=1.3\ut{m/s}\\ \end{cases} $$ $$ \put \begin{cases} \RE : \text{Rotational Kinetic Energy}\\ \KE : \text{Translational Kinetic Energy}\\ \GE : \text{Gravitational Potential Energy}\\ \end{cases} $$ $$ \begin{cases} \Sigma F&=ma\\ \Sigma \tau &= I\alpha\\ a&=r\alpha\\ \end{cases} $$ $$ \begin{cases} T-mg&=m(-a)\\ rT&=I\alpha\\ a&=r\alpha\\ \end{cases} $$ $$ \therefore a=\frac{ m g r^2}{I+m r^2} $$ $$\ab{a}$$ $$S=v_0t+\frac{1}{2}at^2,$$ $$ \begin{aligned} t&=\frac{\sqrt{2 a y+{v_0}^2}-v_0}{a}\\ &\approx 0.8853258140080581\ut{s}\\ &\approx 0.89\ut{s}\\ \end{aligned} $$ $$\ab{b}$$ $$ \omega_0=\frac{v_0}{r},$$ $$\Sigma \Delta E=0,$$ $$\Delta \GE+\Delta \KE+\Delta \RE=0$$ $$\Delta \GE+(\KE_f-\KE_i)+(\RE_f-\RE_i)=0$$ $$ \begin{aligned} \KE_f+\RE_f &=\KE_i+\RE_i-\Delta \GE\\ &=\frac{1}{2}m{v_0}^2+\frac{1}{2}I{\omega_0}^2-mgy\\ &=\frac{1}{2}m{v_0}^2+\frac{1}{2}I\(\frac{v_0}{r}\)^2-mgy\\ &=\frac{18 g}{125}+\frac{10164167}{1280000}\\ &\approx 9.35291306875\ut{J}\\ &\approx 9.4\ut{J}\\ \end{aligned} $$ $$\ab{c}$$ $$2aS=v^2-{v_0}^2,$$ $$ \begin{aligned} v&=\sqrt{2ay+{v_0}^2}\\ &=\sqrt{\frac{2 g m r^2}{I+m r^2}+{v_0}^2}\\ &=\sqrt{\frac{9216 g}{300715}+\frac{169}{100}}\\ &\approx 1.4108663974618418\ut{m/s}\\ &\approx 1.4\ut{m/s}\\ \end{aligned} $$ $$\ab{d}$$ $$ \begin{aligned} \KE&=\frac{1}{2}mv^2\\ &=\frac{1}{2}m(2a+{v_0}^2)\\ &=\frac{1}{2} m \left(\frac{2 m gr^2}{I+m r^2}+{v_0}^2\right)\\ &=\frac{13824 g}{7517875}+\frac{507}{5000}\\ &\approx 0.11943263948921737\ut{J}\\ &\approx 0.12\ut{J}\\ \end{aligned} $$ $$\ab{e}$$ $$ \begin{aligned} \omega&=\frac{v}{r}\\ &=\frac{\sqrt{2a+{v_0}^2}}{r}\\ &=\frac{1}{r}\sqrt{\frac{2 mg y r^2}{I+m r^2}+{v_0}^2}\\ &=\frac{625}{2} \sqrt{\frac{9216 g}{300715}+\frac{169}{100}}\\ &\approx 4.408957492068256\times10^2\ut{rad/s}\\ &\approx 4.4\times10^2\ut{rad/s}\\ \end{aligned} $$ $$\ab{f}$$ $$ \begin{aligned} \RE &=\frac{1}{2}I\omega^2\\ &=\frac{I v^2}{2 r^2}\\ &=\frac{I \left(2 a y+{v_0}^2\right)}{2 r^2}\\ &=\frac{mg y I}{I+m r^2}+\frac{I {v_0}^2}{2 r^2}\\ &=\frac{8550 g}{60143}+\frac{16055}{2048}\\ &\approx 9.233480429260783\ut{J}\\ &\approx 9.2\ut{J}\\ \end{aligned} $$