<1-1> (1.1.1) \frac{dx(x)}{dt}\equiv v(t) <1-2> (1.1.2) s(t)=|v(t)| <1-3> (1.1.3) x(t)=\int v(x)dt+C_1 <1-4> (1.1.4) \begin{equation*} \frac{dv(t)}{dt}\equiv a(t) \end{equation*} <1-5>�@(1.1.5) v(t)=\int a(t)dt+C_2 <2-1> (2.1.1) \vec{a}=\frac{\vec{F}}{m} <2-2> (2.1.2) \vec{F}_{21}=-\vec{F}_{12} \quad\mbox{or}\quad\vec{F}_{21}+\vec{F}_{12}=0 <2-3> (2.1.3) m\frac{d^2x(t)}{dt^2}=F <2-4> (2.2.1) m\frac{d^2x(t)}{dt^2}=0 <2-5>�@(2.2.2) m\frac{dv(t)}{dt}=0 <2-6> v(t)=C_2 <2-7> (2.2.3) v(t)=v_0 <2-8> x(t)=\int v_2dt+C_1 <2-9> (2.2.4) x(t)=v_0t+x_0 <2-10> (2.2.5) m\frac{d^2x(t)}{dt^2}=-mg <2-11> (2.2.6) v(t)=v_0-\int gdt=v_0-gt <2-12> (2.2.7) \begin{array}{ll} x(t)&=\displaystyle{x_0+\int v(t)dt} \\ &=\displaystyle{x_0+\int (v_0-gt)dt} \\ &=\displaystyle{v_0+v_0t-\frac{1}{2}gt^2} \end{array} <2-13> (2.2.8) \left\{\begin{array}{l} \displaystyle{v(t)= -gt} \\ \\ \displaystyle{x(t)=h-\frac{1}{2}gt^2} \end{array}\right. <2-14> m_1\frac{d^2x_1(t)}{dt^2}=F_{21} <2-15> m_2\frac{d^2x_2(t)}{dt^2}=F_{12} <2-16> (2.3.1) F_{12}(x)=-F_{12}(x) <2-17> (2.4.1) m_1\frac{d^2x_1(t)}{dt^2}+ m_2\frac{d^2x_2(t)}{dt^2} =\frac{d^2[m_1x_1(t)+m_2x_2(t)]}{dt^2}=0 <2-18> (2.4.2) X(t)=\frac{ m_1x_1(t)+m_2x_2(t)}{m_1+m_2} <2-19> (2.4.3) Q=\frac{ w_1Q_1+W_2Q_2}{w_1+w_2} <2-20> (2.4.4) M\frac{ d^2X(t) }{dt^2}=0 <2-21> (2.4.5) m\frac{dv(t)}{dt}=F <2-22> (2.4.6) \frac{d[mv(t)]}{dt}=F <2-23> (2.4.7) mv(t)\equiv p(t) <2-24> (2.4.8) \frac{dp(t)}{dt}=F <2-25> (2.4.9) \left\{ \begin{array}{l} \displaystyle{\frac{dp_1(t)}{dt}=F_{21}} \\ \\ \displaystyle{\frac{dp_2(t)}{dt}=F_{12}} \end{array}\right.,, <2-26> (2.4.10) V(t)=\frac{dX(t)}{dt} <2-27> (2.4.11) M\frac{dV(t)}{dt}=0 <2-28> (2.4.12) \frac{dP(t)}{dt}=0 <2-29> \begin{array}{ll} V(t)&=\displaystyle{\frac{d\left(\displaystyle{\frac{m_1x_1(t)+m_2x_2(t)}{m_1+m_2}} \right)}{dt}} \\ \\ &=\displaystyle{\frac{1}{M}\frac{d\left(m_1x_1\right)}{dt} +\frac{1}{M}\frac{d\left(m_2x_2\right)}{dt}} \\ \\ &=\displaystyle{\frac{p_1}{M}+\frac{p_2}{M}} <2-30> (2.4.13) P=p_1+p_2 <2-31> (2.5.1) m\frac{d^2x}{dt^2}=m\frac{dv}{dt}=F(x) <2-32> (2.5.2) \frac{df(y(x))}{dx}=\frac{df(y)}{dy}\frac{dy(x)}{dx} <2-33> (2.5.3) T(v)=\frac{m}{2}v^2 <2-34> \begin{array}{ll} \displaystyle{\frac{dT}{dt}}&=\displaystyle{\frac{dT}{dv}\frac{dv}{dt} }\\ \\ &=\displaystyle{\frac{m}{2}\frac{dv^2}{dv}\frac{dv}{dt}}\\ \\ &=\displaystyle{\frac{m}{2}(2v)\frac{dv}{dt}} \\ \\ &=v\displaystyle{\left(m\frac{dv}{dt}\right)} \end{array} <2-35> \frac{dT}{dt}=F(x)\frac{dx}{dt} <2-36> \int_{t_1}^{t_2}\frac{dT}{dt}dt=\int_{t_1}^{t_2}F(x)\frac{dx}{dt}dt <2-37> \begin{array}{rl} \displaystyle{\int_{t_1}^{t_2}\frac{dT}{dt}}dt&=\displaystyle{\int_{T_1}^{T_2}d} \\ \\ &=T_2-T_1 \end{array} <2-38> \int_{t_1}^{t_2}F(x)\frac{dx}{dt}dt=\int_{x_1}^{x_2}F(x)dx <2-39> (2.5.4) T_2-T_1=\int_{x_1}^{x_2}F(x)dx <2-40> (2.5.5) \int_{x_1}^{x_2}F(x)dx=-\int_{x_2}^{x_1}F(x)dx <2-41> (2.5.6) F(x) =-\frac{dV(x)}{dx} <2-42> (2.5.7) \begin{array}{rl} \displaystyle{\int_{x_1}^{x_2}F(x)dx} &=-\displaystyle{\int_{x_1}^{x_2}\frac{dV(x)}{dx}dx} \\ \\ &=V(x_1)-V(x_2) \end{array} <2-43> (2.5.8) T_1+V(x_1)=T_2+V(x_2) <2-44> \int_{x_0}^{x}F(x)dx=V(x_0)-V(x) <2-45> (2.5.9) V(x)=V(x_0)+\left[-\int_{x_0}^{x}F(x)dx\right] <2-46> \begin{array}{l} V(x_1)-V(x_2) \\ \\ \quad=\displaystyle{\left[V(x_0)+\left\{-\int_{x_0}^{x_1}F(x)dx\right\}\right]- \left[V(x_0)+\left\{-\int_{x_0}^{x_2}F(x)dx\right\}\right]} \\ \\ \quad=\displaystyle{-\int_{x_0}^{x_1}F(x)dx+\int_{x_0}^{x_2}F(x)dx }\\ \\ \quad=\displaystyle{+\int_{x_1}^{x_0}F(x)dx+\int_{x_0}^{x_2}F(x)dx }\\ \\ \quad=\displaystyle{\int_{x_1}^{x_2}F(x)dx} \end{array} <2-47> (2.5.10) V(x)=-\int_{x_0}^{x}F(x)dx <2-48> (2.5.11) \frac{m}{2}v^2+V(x)=E <2-49> (2.6.1) v^2=\frac{2[E-V(x)]}{m} <2-50> (2.6.2) E-V(x)\ge0 <2-51> (2.6.3) \mbox{When}\quad a\ge x\ge b,\quad v^2=\frac{2[E-V(x)]}{m}\ge <2-52> (2.6.4) v=\frac{dx}{dt}=\pm\sqrt{\frac{2}{m}}\sqrt{E-V(x)}} <2-53> (2.6.5) \begin{array}{rl} \displaystyle{\int\frac{1}{\sqrt{E-V(x)}}}dx &=\displaystyle{\sqrt{\frac{2}{m}}\int dt+C} \\ \\ &=\displaystyle{\sqrt{\frac{2}{m}}t+C} \end{array} <2-54> (2.6.6) \left\{ \begin{array}{l} x(t)=x_0+v_0t \\ \\ v(t)=v_0 \end{array} \right. <2-55> (2.6.7) \frac{m}{2}v^2(t)=E <2-56> (2.6.8) E=\frac{m}{2}v_0^2 <2-57> (2.6.9) \int\frac{1}{\sqrt{E}}dx=\pm\sqrt{\frac{2}{m}}t+C <2-58> (2.6.10) \sqrt{\frac{2}{m}}\frac{x}{v_0}=\pm\sqrt{\frac{2}{m}}t+C <2-59> (2.6.11) C=\sqrt{\frac{2}{m}}\frac{x_0}{v_0} <2-60> \sqrt{\frac{2}{m}}\frac{x-x_0}{v_0}=\pm\sqrt{\frac{2}{m}}t <2-61> x-x_0=\pm\sqrt{\frac{2}{m}}t\sqrt{\frac{m}{2}}v_0=\pm v_0t <2-62> (2.6.12) x=x_0\pm v_0t <2-63> v(t)=\frac{dx(t)}{dt}=\pm v_0 <2-64> (2.6.13) v(t)=v_0 <2-65> (2.6.14) \left\{ \begin{array}{l} x(t)=x_0+v_0t \\ \\ v(t)=v_0 \end{array}\right. <2-66> (2.7.1) m\frac{d^2x}{dt^2}=\left\{ \begin{array}{l} -R(v), \quad \mbox{when $v\ge0$ travelling to the right} \\ \\ +R(v), \quad \mbox{when $v\le0$ travelling to the left} \end{array} \right. <2-67> (2.7.2) m\frac{dv}{dt}=-R(v) <2-68> (2.7.3) \begin{array}{rl} m\displaystyle{\int\frac{1}{R(v)}dv}&=-\displaystyle{\int dt+C} \\ \\ &=-t+C \end{array*} <2-69> (2.7.4) R(v)=\left\{ \begin{array}{ll} \alpha v&\mbox{(for small $v$}} \\ \\ \beta v^2&\mbox{(for large $v$)} \end{array}\right. <2-70> \left\{ \begin{array}{ll} m\displaystyle{\int\frac{1}{\alpha v}dv}=\displaystyle{\frac{m}{\alpha}\ln v} =-t+C & \mbox{(for small $v$)} \\ \\ m\displaystyle{\int\frac{1}{\beta v^2}dv}=-\displaystyle{\frac{m}{\beta v}}=-t+C} & \mbox{(for large $v$)} \end{array}\right. <2-71> C=\left\{ \begin{array}{ll} \displaystyle{\frac{m}{\alpha}}\ln v_0 & \mbox{(for small $v$.)} \\ \\ -\displaystyle{\frac{m}{\beta v_0}} & \mbox{(for large $v$.)} \end{array}\right. <2-72> (2.7.5) v(t)=\left\{ \begin{array}{ll} v_0 e^{-\alpha t} & \mbox{(for small $v$.)} \\ \\ -\displaystyle{\frac{v_0}{1+(\beta/m)v_0t}} & \mbox{(for large $v$.)} \end{array}\right. <2-73> (2.7.6) x(t)=\left\{ \begin{array}{ll} v_0\displaystyle{\int e^{-(\alpha/m)t}dt}+C'=-\displaystyle{\frac{mv_0}{\alpha}}e^{-(\alpha/m)t}+C�f & \mbox{(for small $v$.)} \\ \\ v_0\displaystyle{\int\frac{1}{1+(\beta/m)v_0t}}+C�f =\frac{\beta}{m}\ln\left(1+\frac{v_0\beta}{m}t\right)+C�f & \mbox{(for large $v$.)} \end{array}\right. <2-74> (2.7.7) C�f=\left\{ \begin{array}{ll} x_0+\displaystyle{\frac{mv_0}{\alpha}} & \mbox{(for small $v$.)} \\ \\ x_0 & \mbox{(for large $v$.)} \end{array}\right. <2-75> (2.7.8) x(t)=\left\{ \begin{array}{ll} x_0+\displaystyle{\frac{mv_0}{\alpha}}\left[1-e^{-(\alpha/m)t} \right] & \mbox{(for small $v$.)} \\ \\ x_0+\displaystyle{\frac{\beta}{m}}\ln\left(1+\frac{v_0\beta}{m}t\right) & \mbox{(for large $v$.)} \end{array}\right. <2-76> x(t)\rightarrow\left\{ \begin{array}{ll} x_0+\displaystyle{\frac{mv_0}{\alpha}}& \mbox{(for small $v$.)} \\ \\ \infty & \mbox{(for large $v$.)} \end{array}\right. <3-1> (3.1.1) E=\frac{m}{2}v^2+V(x) <3-2> (3.1.2) F(x)=-\frac{dV(x)}{dx} <3-3> (3.1.3) F(x)=-kx <3-4> (3.1.4) m\frac{d^2x}{dt^2}=-kx <3-5> (3.1.5) \frac{d^2x}{dt^2}=-\omega^2x <3-6> (3.1.6) x(t)=A\sin(\omega t+B) <3-7> (3.1.7) \frac{dx}{dt}=A\omega\cos(\omega t+B) <3-8> (3.1.8) \left\{ \begin{array}{l} a=A \sin B \\ \\ 0=A\omega\cos B \end{array} \right. <3-9> (3.1.9) A=a,\quad B=\frac{\pi}{2} <3-10> (3.1.10) \left\{ \begin{array}{l} x(t)=a \sin\left(\omega t+\displaystyle{\frac{\pi}{2}}\right)=a\cos(\omega t) \\ \\ v(t)=a \omega\cos\left(\omega t+\displaystyle{\frac{\pi}{2}}\right)=-a\omega\sin(\omega t) \end{array} \right. <3-11> (3.1.11) V(x)=-\int F(x)dx <3-12> (3.1.12) V(x)=-\int (-kx)dx=\frac{1}{2}kx^2 <3-13> (3.1.13) F(x_0)=-V^{(1)}(x_0)\equiv V^\prime(x_0)=0 <3-14> (3.1.14) \begin{array}{rl} V(x)=& V(x_0)+\displaystyle{\frac{V^{(1)}(x_0)}{1!}}(x-x_0)+ \displaystyle{\frac{V^{(2)}(x_0)}{2!}}(x-x_0)^2 \\ \\ & +\displaystyle{\frac{V^{(3)}(x_0)}{3!}}(x-x_0)^3+\cdots \end{array} <3-15> (3.1.15) V(x)=V(x_0)+ \frac{V^{(2)}(x_0)}{2!}(x-x_0)^2 <3-16> (3.1.16) V^{(2)}(x_0)\equiv k\quad\mbox{($k$ is aconstant.)} <3-17> (3.1.17) V(x)=V(0)+\frac{1}{2}kx^2 <3-18> (3.2.1) m\frac{d^2x}{dt^2}=-kx-\alpha\frac{dx}{dt} <3-19> (3.2.2) \left\{ \begin{array}{l} \sqrt{\displaystyle{\frac{k}{m}}}=\omega \\ \\ \displaystyle{\frac{\alpha}{m}}=2\gamma \end{array} \right. <3-20> (3.2.3) \frac{d^2x}{dt^2}+2\gamma\frac{dx}{dt}+\omega^2x=0 <3-21> (3.2.4) x(t)=C_1e^{\left(-\gamma+\sqrt{\gamma^2-\omega^2}\right)t}+ C_2e^{\left(-\gamma-\sqrt{\gamma^2-\omega^2}\right)t} <3-22> (3.2.5) \begin{array}{ll} \displaystyle{\frac{dx(t)}{dt}}= & \left(-\gamma+\sqrt{\gamma^2-\omega^2}\right)C_1 e^{\left(-\gamma+\sqrt{\gamma^2-\omega^2}\right)t} \\ \\ & +\left(-\gamma-\sqrt{\gamma^2-\omega^2}\right)C_2 e^{\left(-\gamma-\sqrt{\gamma^2-\omega^2}\right)t} \end{array} <3-23> (3.2.6) \left\{ \begin{array}{l} C_1+C_2=a \\ \\ \left(-\gamma+\sqrt{\gamma^2-\omega^2}\right)C_1 +\left(-\gamma-\sqrt{\gamma^2-\omega^2}\right)C_2=0 \end{array}\right. <3-24> (3.2.7) \left\{ \begin{array}{l} C_1=\displaystyle{\frac{1}{2}}\left(1-\displaystyle{ \frac{\gamma}{\sqrt{\gamma^2-\omega^2}}}\right)a \\ \\ C_2=\displaystyle{\frac{1}{2}}\left(1+\displaystyle{ \frac{\gamma}{\sqrt{\gamma^2-\omega^2}}}\right)a \end{array}\right. <3-25> (3.2.8) \begin{array}{rl} x(t)=\displaystyle{\frac{a}{2}}& \left[ \left(1-\displaystyle{\frac{\gamma}{\sqrt{\gamma^2-\omega^2}}} e^{\left(-\gamma+\sqrt{\gamma^2-\omega^2}\right)t} \right)\right.\\ \\ & +\left.\left(1+\displaystyle{\frac{\gamma}{\sqrt{\gamma^2-\omega^2}}} e^{\left(-\gamma-\sqrt{\gamma^2-\omega^2}\right)t}\right)\right] \end{array} <3-26> (3.2.9) \begin{array}{rl} x(t)=\displaystyle{\frac{a}{2}}& \left[ \left(1-\displaystyle{\frac{\gamma}{\sqrt{\gamma^2-\omega^2}}} e^{\left(-\gamma+\sqrt{\gamma^2-\omega^2}\right)t} \right)\right.\\ \\ & +\left.\left(1+\displaystyle{\frac{\gamma}{\sqrt{\gamma^2-\omega^2}}} e^{\left(-\gamma-\sqrt{\gamma^2-\omega^2}\right)t}\right)\right] \end{array} <3-27> (3.2.10) \begin{array}{rl} x(t)=\displaystyle{\frac{a}{2}}e^{-\gamma t}& \left[ \left(1+\displaystyle{\frac{i\gamma}{\sqrt{\omega^2-\gamma^2}}} e^{\left(i\sqrt{\omega^2-\gamma^2}\right)t} \right)\right.\\ \\ & +\left.\left(1-\displaystyle{\frac{i\gamma}{\sqrt{\omega^2-\gamma^2}}} e^{\left(-i\sqrt{\omega^2-\gamma^2}\right)t}\right)\right] \end{array} <3-28> (3.2.11) \begin{array}{rl} e^{\pm i\sqrt{\omega^2-\gamma^2}t}= & \cos\left(\sqrt{\omega^2-\gamma^2}t\right) \\ \\ & \quad \pm i\sin\left(\sqrt{\omega^2-\gamma^2}t\right) \end{array} <3-29> (3.2.12) \begin{array}{rl} x(t)= ae^{-\gamma t}& \left[\cos\left(\sqrt{\omega^2-\gamma^2}t\right) \right. \\ \\ & \quad\left.-\displaystyle{\frac{\gamma}{\sqrt{\omega^2-\gamma^2}} \sin\left(\sqrt{\omega^2-\gamma^2}t\right)\right] \end{array} <3-30> (3.3.1) m\frac{d^2x}{dt^2}=-kx+F\cos(\Omega t) <3-31> (3.3.2) \frac{d^2x}{dt^2}+\omega^2x=f\cos(\Omega t) <3-32> (3.3.3) x(t)=C_1e^{i\omega t}+C_2 e^{-i\omega t}+ \frac{f}{\left|\Omega^2-\omega^2\right|} \cos(\Omega t) <3-33> (3.3.4) \begin{array}{rl} \displaystyle{\frac{dx}{dt}}=i\omega & \left(C_1e^{i\omega t}-C_2 e^{-i\omega t}\right) \\ \\ & -\displaystyle{\frac{f\Omega}{\left|\Omega^2-\omega^2\right|} }\sin(\Omega t) \end{array} <3-34> (3.3.5) i\omega\left(C_1-C_2 \right)=0 <3-35> (3.3.6) x(t)=2C_1\cos(\omega t)+\frac{f}{\left|\Omega^2-\omega^2\right|} \cos(\Omega t) <3-36> C_1=\frac{1}{2}\left(a-\frac{f}{\left|\Omega^2-\omega^2\right|}\right) <3-37> \cos A-\cos B=-2\cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) <3-38> (3.3.7) \begin{array}{rl} x(t)= & a\cos(\omega t) \\ \\ & \quad-\displaystyle{\frac{2f}{\left|\Omega^2-\omega^2\right| \sin\left[\left(\displaystyle\frac{\Omega+\omega}{2}\right)t\right] \sin\left[\left(\displaystyle\frac{\Omega-\omega}{2}\right)t\right]} \end{array} <3-39> (3.4.1) m\frac{d^2x}{dt^2}=-kx-\alpha\frac{dx}{dt}+F\cos(\Omega t) <3-40> (3.4.2) \frac{d^2x}{dt^2}+\gamma\frac{dx}{dt}+\omega^2x=f\cos(\Omega t) <3-41> (3.4.3) \begin{array}{rl} x(t)= & e^{-\gamma t}\left[C_1e^{\sqrt{\gamma^2-\omega^2}t}+ C_2e^{-\sqrt{\gamma^2-\omega^2}t}\right] \\ \\ & \quad+f\cos\left(\Omega t-\phi\right) \end{array} <3-42> (3.4.4) \tan\phi=\displaystyle{\frac{2\gamma\Omega}{\omega^2-\Omega^2}} <3-43> \frac{m}{2}v^2+V(x)=\frac{m}{2}a^2\omega^2\equiv E <4-1> (4.1.1) \begin{equation*} \vec{r}=\vec{i}x+\vec{j}y \end{equation*} <4-2> (4.1.2) \vec{r}=\vec{i}x+\vec{j}y+\vec{k}z <4-3> (4.1.3) \vec{r}=\vec{i}x+\vec{j}y <4-4> (4.1.4) \left\{ \begin{array}{rl} & x=r\cos\theta \\ \\ & y=r\sin\theta \end{array} \right. <4-5> (4.1.5) \left\{ \begin{array}{l} r=\sqrt{x^2+y^2} \\ \\ \theta=\tan^{-1}\left(\displaystyle{\frac{y}{x}}\right) \end{array} \right. <4-6> (4.1.6) \rho=r\cos\left(\frac{\pi}{2}-\theta\right)=r\sin\theta <4-7> (4.1.7) \left\{ \begin{array}{l} x=\rho\cos\phi \\ \\ y=\rho\sin\phi \end{array} \right. <4-8> (4.1.8) \left\{ \begin{array}{l} x=r\sin\theta\cos\phi \\ \\ y=r\sin\theta\sin\phi \end{array} \right. <4-9> (4.1.9) z=r\cos\theta <4-10> (4.1.10) \left\{ \begin{array}{l} x=r\sin\theta\cos\phi \\ \\ y=r\sin\theta\sin\phi \\ \\ z=r\cos\theta \end{array} \right.,\quad (-\pi\le\theta<\pi, 0\le\phi<2\pi) <4-11> (4.1.11) \left\{ \begin{array}{l} r=\sqrt{x^2+y^2+z^2} \\ \\ \theta=\tan^{-1}\left(\displaystyle{\frac{x^2+y^2}{z}}\right) \\ \\ \phi=\tan^{-1}\left(\displaystyle{\frac{y}{x}}\right) \end{array} \right. <4-12> \vec{F}=\vec{i}F_x+\vec{j}F_y+\vec{k}F_z <4-13> (4.2.1) \left. \begin{array}{l} \displaystyle{m\frac{d^2x(t)}{dt^2}=F_x} \\ \\ \displaystyle{m\frac{d^2y(t)}{dt^2}=F_y} \\ \\ \displaystyle{m\frac{d^2z(t)}{dt^2}=F_z} \end{array}\right\} \quad \rightarrow\quad \displaystyle{m\frac{d^2\vec{r}(t)}{dt^2}=\vec{F}} <4-14> (4.2.2) \left\{ \begin{array}{l} \vec{r}(0)\equiv\vec{r}_0=\vec{i}0+\vec{j}h+\vec{k}0 \\ \\ \vec{v}(0)\equiv\vec{v}_0=\vec{i}v_0\cos\theta+\vec{j}v_0\sin\theta+\vec{k}0 \end{array}\right. <4-15> \vec{r}(t)=\vec{i}x(t)+\vec{j}y(t)+\vec{k}z(t) <4-16> (4.2.3) m\frac{d^2\vec{r}(t)}{dt^2}=\vec{F} <4-17> (4.2.4) \vec{F}=-\vec{j}mg <4-18> (4.2.5) \left\{ \begin{array}{l} m\displaystyle{\frac{d^2x(t)}{dt^2}}=0 \\ \\ m\displaystyle{\frac{d^2y(t)}{dt^2}}=-mg\\ \\ m\displaystyle{\frac{d^2z(t)}{dt^2}}=0 \end{array}\right. <4-19> (4.2.6) \left\{ \begin{array}{l} \left\{\begin{array}{l} x(t)=(v_0\cos\theta)t \\ \\ v_x(t)=v_0\cos\theta \end{array}\right. \\ \\ \left\{\begin{array}{l} y(t)=h+(v_0\sin\theta)t-\displaystyle{\frac{1}{2}}gt^2 \\ \\ v_y(t)=v_0\sins\theta-gt \end{array}\right. \\ \\ \left\{\begin{array}{l} z(t)=0 \\ \\ v_z(t)=0 \end{array}\right. \end{array}\right. <4-20> (4.2.7) \left\{ \begin{array}{l} x(t)=(v_0\cos\theta)t \\ \\ y(t)=h+(v_0\sin\theta)t-\displaystyle{\frac{1}{2}}gt^2 \end{array}\right. <4-21> t=\frac{v_0\sin\theta+\sqrt{v_0^2\sin^2\theta+2gh}}{g} <4.22> (4.2.8) y=-\frac{g}{2v_0^2\cos^2\theta}(x-X)^2+\frac{v_0^2\sin^2\theta}{2g} <4-23> X=\frac{v_0^2\sin(2\theta)}{2g} <4-24> y=\frac{v_0^2\sin^2\theta}{2g} <4-25> \left(\frac{dr}{dt},\frac{d\theta}{dt}\equiv\omega\right) <4-26> (4.3.1) \left\{ \begin{array}{l} \displaystyle{\frac{dx}{dt}=\frac{dr}{dt}\cos\theta-r\omega\sin\theta} \\ \\ \displaystyle{\frac{dy}{dt}=\frac{dr}{dt}\sin\theta+r\omega\cos\theta} \end{array}\right. <4-27> \frac{df(x(t))}{dt}=\frac{df(x)}{dx}\frac{dx(t)}{dt} <4-28> \frac{dx}{dt}\cos\theta+\frac{dy}{dt}\sin\theta=\frac{dr}{dt} <4-29> -\frac{dx}{dt}\sin\theta+\frac{dy}{dt}\cos\theta=r\omega <4-30> (4.3.2) \left\{ \begin{array}{l} \displaystyle{\frac{dr}{dt}=\frac{dx}{dt}\cos\theta+\frac{dy}{dt}\sin\theta} \\ \\ r\displaystyle{\frac{d\theta}{dt}=-\frac{dx}{dt}\sin\theta+\frac{dy}{dt}\cos\theta} \end{array}\right. <4-31> (4.3.3) \left\{ \begin{array}{l} \displaystyle{\frac{d^2x}{dt^2}=\left(\frac{d^2r}{dt^2}-r\omega^2\right)\cos\theta -\left(r\frac{d\omega}{dt}+2\frac{dr}{dt}\omega\right)\sin\theta} \\ \\ \displaystyle{\frac{d^2y}{dt^2}=\left(\frac{d^2r}{dt^2}-r\omega^2\right)\sin\theta +\left(r\frac{d\omega}{dt}+2\frac{dr}{dt}\omega\right)\cos\theta} \end{array}\right. <4-32> \frac{d^2x}{dt^2}\cos\theta+\frac{d^2y}{dt^2}\sin\theta =\frac{d^2r}{dt^2}-r\omega^2 <4-33> -\frac{d^2x}{dt^2}\sin\theta+\frac{d^2y}{dt^2}\cos\theta =r\frac{d\omega}{dt}+2\frac{dr}{dt}\omega <4-34> \frac{d\{f(t)g(t)\}}{dt}=\frac{df(t)}{dt}g(t)+f(t)\frac{dg(t)}{dt} <4-35> \frac{d(r^2\omega)}{dt}=2r\frac{dr}{dt}\omega+r^2\frac{d\omega}{dt} <4-36> \left\{ \begin{array}{l} \displaystyle{\frac{d^2r}{dt^2}-r\omega^2=\frac{d^2x}{dt^2}\cos\theta+\frac{d^2y}{dt^2}\sin\theta} \\ \\ \displaystyle{\frac{1}{r}\frac{d(r^2\omega)}{dt}=-\frac{d^2x}{dt^2}\sin\theta+\frac{d^2y}{dt^2}\cos\theta} \end{array}\right. <4-37> (4.3.5) \left\{ \begin{array}{l} \displaystyle{m\frac{d^2x}{dt^2}=F_x} \\ \\ \displaystyle{m\frac{d^2y}{dt^2}=F_y} \end{array}\right. <4-38> (4.3.6) \left\{ \begin{array}{l} \displaystyle{m\left(\frac{d^2r}{dt^2}-r\omega^2\right) =F_x\cos\theta+F_y\sin\theta} \\ \\ \displaystyle{m\frac{1}{r}\frac{d(r^2\omega)}{dt} =-F_x\sin\theta+F_y\cos\theta} \end{array}\right. <4-39> (4.3.7) \vec{F}=\frac{\vec{r}}{r}f. <4-40> (4.3.8) \vec{F}=-\frac{\vec{r}}{r}F <4-41> \frac{\vec{r}}{r}=\vec{i}\cos\theta+\vec{j}\sin\theta <4-42> \vec{F}=-\vec{i}F\cos\theta-\vec{j}F\sin\theta <4-43> (4.3.9) \left\{ \begin{array}{l} F_x=-F\cos\theta \\ \\ F_y=-F\sin\theta \end{array}\right. <4-44> (4.3.10) \left\{ \begin{array}{l} \displaystyle{m\left(\frac{d^2r}{dt^2}-r\omega^2\right)=-F }\\ \\ \displaystyle{\frac{m}{r}\frac{d(r^2\omega)}{dt}=0} \end{array}\right. <4-45> (4.3.11) m\frac{d^2r}{dt^2}=-F+\frac{mC^2}{r^3} <4-46> (4.3.12) F=\frac{mC^2}{\ell ^3} <4-47> r^2\omega=\ell ^2\omega=C <4-48> (4.3.13) \omega=\frac{C}{\ell ^2}=\mbox{constant}\equiv\omega_0 <4-49> (4.4.1) \vec{F}=\vec{j}mg <4-50> (4.4.2) \vec{S}=-\frac{\vec{r}}{r}S <4-51> (4.4.3) m\frac{d^2\vec{r}}{dt^2}=\vec{F}+\vec{S} <4,52> (4.4.4) \left\{ \begin{array}{l} \displaystyle{m\frac{d^2x}{dt^2}=-\frac{x}{r}S }\\ \\ \displaystyle{m\frac{d^2y}{dt^2}=mg-\frac{y}{r}S }\\ \\ \displaystyle{m\frac{d^2z}{dt^2}=0 \end{array}\right. <4.53> (4.4.5) \left\{ \begin{array}{l} x=\ell\sin\phi\\ \\ y=\ell\cos\phi \end{array}\right. <4-54> (4.4.6) \left\{ \begin{array}{l} x=\ell \phi\\ \\ y=\ell \end{array}\right. <4-55> (4.4.7) \left\{ \begin{array}{l} \displaystyle{m\frac{d^2x}{dt^2}=-\frac{x}{\ell}S }\\ \\ 0=mg-S \end{array}\right. <4-56> \frac{d^2x}{dt^2}=-\frac{g}{\ell}x <4-57> (4.4.8) \omega=\sqrt{\frac{g}{\ell}} <4-58> (4.4.9) \frac{d^2x}{dt^2}=-\omega^2x <4-59> (4.4.10) x(t)=A\sin(\omega t+B) <4-60> (4.4.11) x(t)=a\cos(\omega t) <4-61> (4.4.12) \left\{ \begin{array}{l} \omega=\displaystyle{\sqrt{\frac{g}{\ell}}}\\ \\ T=2\pi \displaystyle{\sqrt{\frac{\ell}{g}}} \end{array}\right. <4-62> (4.4.13) m\frac{d^2\vec{r}}{dt^2}=-k\vec{r}\quad\Rightarrow\quad\left\{ \begin{array}{l} \displaystyle{m\frac{d^2x}{dt^2}=-kx} \\ \\ \displaystyle{m\frac{d^2y}{dt^2}=-ky} \end{array}\right. <4-63> (4.4.14) \left\{ \begin{array}{l} x(t)=a_1\sin(\omega t+b_1) \\ \\ y(t)=a_2\sin(\omega t+b_2) \end{array}\right. <4-64> \sin(A+B)=\sin A\cos B+\cos A\sin B <4-65> \sin\omega t=\frac{1}{\sin (b_2-b_1)}\left(\frac{x}{a_1}\sin b_2 -\frac{y}{a_2}\sin b_1\right) <4-66> \cos\omega t=\frac{1}{\sin (b_2-b_1)}\left(-\frac{x}{a_1}\cos b_2 +\frac{y}{a_2}\cos b_1\right) <4-67> \cos(A+B)=\cos A\cos B-\sin A\sin B <4-68> (4.4.15) \sin^2(b_2-b_1)=\frac{x^2}{a_1^2}+\frac{y^2}{a_2^2} -2\frac{x}{a_1}\frac{y}{a_2}\cos(b_2-b_1) <4-69> (4.4.16) 1=\frac{x^2}{a_1^2}+\frac{y^2}{a_2^2} <4-70> (4.5.1) m\frac{d^2\vec{r}(t)}{dt^2}=\vec{F}(x,y,z) <4-71> (4.5.2) m\frac{d\vec{v}}{dt}=\vec{F}(x,y,z) <4-72> (4.5.3) m\left(\vec{v}\cdot\frac{d\vec{v}}{dt}\right)=\vec{v}\cdot\vec{F}(x,y,z) <4-73> (4.5.4) \vec{v}\cdot\vec{v}=v_x^2+ v_y^2+ v_z^2\equiv v^2 <4-74> (4.5.5) \begin{array}{rl} \displaystyle{\frac{d(v^2)}{dt}} &=\displaystyle{\frac{d(v_x^2+ v_y^2+ v_z^2)}{dt}} \\ \\ &=2\displaystyle{\left(v_x\frac{dv_x}{dt}+ v_y\frac{dv_y}{dt}+ v_z\frac{dv_z}{dt}\right) }\\ \\ &=2\displaystyle{\left(\vec{v}\cdot\frac{d\vec{v}}{dt\right)} \end{array}\right. <4-75> (4.5.6) \frac{m}{2}\frac{dv^2}{dt}=\frac{d\left(m\vec{v}^2/2\right)}{dt}} =\vec{v}\cdot\vec{F}(x,y,z) <4-76> (4.5.7) \int_{t_1}^{t_2}\frac{dT}{dt}dt=\int_{t_1}^{t_2}\left\{\vec{v}\cdot\vec{F}(x,y,z)\right\}dt <4-77> (4.5.8) \int_{T_1}^{T_2}dT=T_2-T_1 <4-78> (4.5.9) \begin{array}{rl} \displaystyle{\int_{t_1}^{t_2}\vec{v}\cdot\vec{F}(x,y,z)dt} &=\displaystyle{\int_{t_1}^{t_2}\frac{dx}{dt}F_xdt+\int_{t_1}^{t_2}\frac{dy}{dt}F_ydt +\int_{t_1}^{t_2}\frac{dz}{dt}F_zdt} \\ \\ &=\displaystyle{\int_{x_1}^{x_2}F_xdx+\int_{y_1}^{y_2}F_ydy +\int_{z_1}^{z_2}F_zdz} \\ \\ &\equiv\displaystyle{\int_{P_1}^{P_2}\left(F_xdx+F_ydy+F_zdz\right)} \end{array} <4-79> (4.5.10) T_2-T_1=\int_{P_1}^{P_2}\left(F_xdx+F_ydy+F_zdz\right) <4-80> (4.5.11) \left.\begin{array}{l} F_x\equiv\displaystyle{-\frac{\partial V(x,y,z)}{\partial x} \\ \\ F_y\equiv\displaystyle{-\frac{\partial V(x,y,z)}{\partial y} \\ \\ F_z\equiv\displaystyle{-\frac{\partial V(x,y,z)}{\partial z} \end{array}\right\}\quad\rightarrow\quad \vec{F}=-\mbox{grad}V(x,y,z) <4-81> (4.5.12) \int_{P_1}^{P_2}\vec{F}\cdot d\vec{S}=-\int_{P_1}^{P_2}\mbox{grad}V(x,y,z)\cdot d\vec{S} <4-82> d\vec{S}=\vec{i}dx+\vec{j}dy+\vec{k}dz <4-83> (4.5.13) \mbox{grad}V(x,y,z)\cdot d\vec{S}=\frac{\partial V}{\partial x}dx +\frac{\partial V}{\partial y}dy+\frac{\partial V}{\partial z}dz <4-84> (4.5.14) \begin{array}{rl} dV(x,y,z)&=V(x+dx,y+dy,z+dz)-V(x,y,z) \\ \\ &=\displaystyle{\frac{\partial V}{\partial x}dx +\frac{\partial V}{\partial y}dy+\frac{\partial V}{\partial z}dz} \end{array} <4-85> (4.5.15) \int_{P_1}^{P_2}\vec{F}\cdot d\vec{S} =-\int_{P_1}^{P_2}dV=V_1-V_2 <4-86> T_2-T_1=V_1-V_2 <4-87> (4.5.16) T_1+V_1=T_2+V_2 <4-88> (4.6.1) \vec{L}(t)= \vec{r}(t)\times\vec{p}(t) <4-89> (4.6.2) \left|\vec{L}\right|= \left|\vec{r}\right|\left|\vec{p}\right|\sin\theta <4-90> (4.6.3) \vec{r}(t)= \vec{i}x+ \vec{j}y+ \vec{k}z <4-91> (4.6.4) \vec{F}(r)= \vec{e}_rf(r) <4-92> (4.6.5) \vec{e}_r= \frac{1}{r}\vec{r}=\frac{\vec{r}}{r} <4-93> \left(\vec{e}_r\cdot\vec{e}_r \right) <4-94> (4.6.6) \left\{\begin{array}{l} x=r\sin\theta\cos\phi \\ \\ y=r\sin\theta\sin\phi \\ \\ z=r\cos\theta \end{array}\right.,\quad\left\{\begin{array}{l} r=\displaystyle{\sqrt{x^2+y^2+z^2}} \\ \\ \theta=\displaystyle{\tan^{-1}\frac{\sqrt{x^2+y^2}}{z}} \\ \\ \phi=\displaystyle{\tan^{-1}\frac{y}{x}} \end{array}\right. <4-95> (4.6.7) \vec{r}\times\vec{F}(r)=0 <4-96> (4.6.8) m\frac{d^2\vec{r}}{dt^2}=\vec{F} <4-97> (4.6.9) m\frac{d\vec{v}}{dt}=\vec{F} <4-98> (4.6.10) m\left(\vec{r}\times\frac{d\vec{v}}{dt}\right) = \vec{r}\times \vec{F}=0 <4-99> \begin{array}{rl} \displaystyle{\frac{d\left(\vec{r}\times\vec{v}\right)}{dt}} &=\displaystyle{\frac{d\vec{r}}{dt}\times\vec{v}+{\vec{r}\times\frac{d\vec{v}}{dt}} \\ \\ &= \displaystyle{\vec{v}\times\vec{v}+{\vec{r}\times\frac{d\vec{v}}{dt}} \\ \\ &=\displaystyle{{\vec{r}\times\frac{d\vec{v}}{dt}} \end{array} <4-100> (4.6.11) \begin{array}{rl} \displaystyle{m\left(\vec{r}\times \frac{d\vec{v}}{dt}\right)} &=\displaystyle{\frac{d\left(\vec{r}\times\vec{p}\right)}{dt}} \\ \\ &= \displaystyle{ \frac{d\vec{L}}{dt}} \\ \\ &=0 \end{array} <4-101> (4.6.12) m\frac{d^2\vec{r}}{dt^2}=\vec{F}\quad\rightarrow\quad\left\{ \begin{array}{l} \displaystyle{m\frac{d^2x}{dt^2}}=F_x \\ \\ \displaystyle{m\frac{d^2y}{dt^2}}=F_y \end{array} <4-102> (4.6.13) \left\{ \begin{array}{l} \displaystyle{\frac{dx}{dt}=\frac{dr}{dt}\cos\theta-r\frac{d\theta}{dt}\sin\theta \\ \\ \displaystyle{\frac{dy}{dt}=\frac{dr}{dt}\sin\theta+r\frac{d\theta}{dt}\cos\theta \end{array}\right. <4-103> (4.6.14) \left\{ \begin{array}{l} \displaystyle{\frac{d^2x}{dt^2}=\left(\frac{d^2r}{dt^2}-r\omega^2\right)\cos\theta -\left(2\frac{dr}{dt}\omega+r\frac{d\omega}{dt}\right)\sin\theta \\ \\ \displaystyle{\frac{d^2y}{dt^2}=\left(\frac{d^2r}{dt^2}-r\omega^2\right)\sin\theta +\left(2\frac{dr}{dt}\omega+r\frac{d\omega}{dt}\right)\cos\theta \end{array}\right. <4-104> \left\{ \begin{array}{l} \displaystyle{m\left(\frac{d^2r}{dt^2}-r\omega^2\right)\cos\theta -m\left(2\frac{dr}{dt}\omega+r\frac{d\omega}{dt}\right)\sin\theta=F_x} \\ \\ \displaystyle{m\left(\frac{d^2r}{dt^2}-r\omega^2\right)\sin\theta +m\left(2\frac{dr}{dt}\omega+r\frac{d\omega}{dt}\right)\cos\theta =F_y} \end{array}\right. <4-105> (4.6.15) \left\{ \begin{array}{l} \displaystyle{m\left(\frac{d^2r}{dt^2}-r\omega^2\right) =F_x\cos\theta+F_y\sin\theta} \\ \\ \displaystyle{m\left(2\frac{dr}{dt}\omega+r\frac{d\omega}{dt}\right) =-F_x\sin\theta+F_y\cos\theta } \end{array}\right. <4-106> (4.6.16) \left\{ \begin{array}{l} \displaystyle{F_x\cos\theta+F_y\sin\theta=F_r} \\ \\ \displaystyle{-F_x\sin\theta+F_y\cos\theta=F_\theta } \end{array}\right. <4-107> (4.6.17) m\left(\frac{d^2r}{dt^2}-r\omega^2\right)=F_r <4-108> (4.6.18) m\left(2\frac{dr}{dt}\omega+r\frac{d\omega}{dt}\right)=F_\theta <4-109> (4.6.19) \left\{ \begin{array}{l} \displaystyle{F_x=\frac{x}{r}f(r)=f(r)\cos\theta} \\ \\ \displaystyle{F_y=\frac{y}{r}f(r)=f(r)\sin\theta} \end{array}\right. <4-110> (4.6.20) \left\{ \begin{array}{rl} F_r&=F_x\cos\theta+F_y\sin\theta \\ \\ &=f(r)\cos^2\theta+f(r)\sin^2\theta \\ \\ &=f(r) \\ \\ F_\theta&=-F_x\sin\theta+F_y\cos\theta \\ \\ &=-f(r)\cos\theta\sin\theta+f(r)\sin\theta\cos\theta \\ \\ &=0 \end{array}\right. <4-111> (4.6.21) m\left(\frac{d^2r}{dt^2}-r\omega^2\right)=f(r) <4-112> (4.6.22) m\left(2\frac{dr}{dt}\omega+r\frac{d\omega}{dt}\right)=0 <4-113> \frac{d\left(f(t)g(t)\right)}{dt}=\frac{df(t) }{dt}g(t)+f(t)\frac{dg(t) }{dt} <4-114> \begin{array}{rl} \displaystyle{\frac{d(r^2\omega)}{dt}}&=\displaystyle{2r\frac{dr}{dt}\omega+r^2\frac{d\omega}{dt}} \\ \\ &=\displaystyle{r\left(2\frac{dr}{dt}\omega+r\frac{d\omega}{dt}\right)} \end{array}\right. <4-115> (4.6.23) \frac{m}{r}\frac{d(r^2\omega)}{dt}=0 <4-116> (4.6.24) mr^2\omega=h <4-117> \vec{a}\times\vec{b}=\vec{i} (a_yb_z-a_zb_y) +\vec{j} (a_zb_x-a_xb_z)+\vec{k} (a_xb_y-a_yb_x) <4-118> (4.6.25) L_z=xp_y-yp_x=m\left(x\frac{dy}{dt}-y\frac{dx}{dt}\right) <4-119> (4.6.26) \begin{array}{rl} L_z&=\displaystyle{mr\cos\theta\left(\frac{dr}{dt}\sin\theta+r\omega\cos\theta\right) - mr\sin\theta\left(\frac{dr}{dt}\cos\theta-r\omega\sin\theta\right)} \\ \\ &=mr^2\omega \end{array} <4-120> \begin{array}{rl} \displaystyle{\frac{dA}{dt}} &=\displaystyle{ \frac{d\left(\displaystyle{\frac{r^2\theta}{2}}\right)}{dt}} \\ \\ &=\displaystyle{\frac{r^2}{2}\frac{d\theta}{dt}} \\ \\ &=\displaystyle{\frac{r^2}{2}\omega} \end{array} <4-121> (4.6.27) L_z=2m\frac{dA}{dt} <4-122> (4.7.1) \vec{F}=\frac{\vec{r}}{r}\frac{K}{r^2} <4-123> (4.7.2) \left\{\begin{array}{l} F_x=\displaystyle{ K\frac{\cos\theta}{r^2}} \\ \\ F_y=\displaystyle{ K\frac{\sin\theta}{r^2}} \\ \\ \end{array}\right. <4-124> (4.7.3) \left\{\begin{array}{rl} F_r&=F_x\cos\theta+F_y\sin\theta \\ \\ &=\displaystyle{ K\frac{\cos^2\theta}{r^2}+ K\frac{\sin^2\theta}{r^2}} \\ \\ &=\displaystyle{ \frac{K}{r^2}} \\ \\ F_\theta&=- F_x\sin\theta+F_y\cos\theta \\ \\ &=\displaystyle{-K\frac{\cos\theta}{r^2}\sin\theta+ K\frac{\sin\theta}{r^2}\cos\theta} \\ \\ &=0 \end{array}\right. <4-125> (4.7.4) m\left(\frac{d^2r}{dt^2}-r\omega^2\right)= \frac{K}{r^2} <4-126> (4.7.5) m\left(2\frac{dr}{dt}\omega+r\frac{d\omega}{dt}\right)=0 <4-127> (4.7.6) m\frac{dv}{dt}-\frac{h^2}{mr^3}-\frac{K}{r^2}=0 <4-128> (4.7.7) m\frac{dv}{dt}=\frac{h^2}{mr^3}+\frac{K}{r^2} <4-129> (4.7.8) mv\frac{dv}{dt}-\frac{h^2}{mr^3}v-\frac{K}{r^2}v=0 <4-130> \frac{d\left(\displaystyle{\frac{m}{2}}v^2\right)}{dt}=mv\frac{dv}{dt} <4-131> \begin{array}{rl} \displaystyle{\frac{d\left(\displaystyle{\frac{h^2}{2mr^2}}\right)}{dt}} &=\displaystyle{\frac{d\left(\displaystyle{\frac{h^2}{2mr^2}}\right)}{dr}\frac{dr}{dt}\right)}\\ \\ &=-\displaystyle{\frac{h^2}{mr^3}}v \end{array} <4-132> \begin{array}{rl} \displaystyle{\frac{d\left(\displaystyle{-\frac{K}{r}}\right)}{dt}} &=\displaystyle{\frac{d\left(\displaystyle{-\frac{K}{r}}\right)}{dr} \frac{dr}{dt}\right)}\\ \\ &=\displaystyle{\frac{K}{r^2}}v \end{array} <4-133> (4.7.9) \frac{d}{dt}\left(\frac{m}{2}}v^2+\frac{h^2}{2mr^2}}+\frac{K}{r}\right)=0 <4-134> (4.7.10) \frac{m}{2}v^2+\frac{h^2}{2mr^2}}+\frac{K}{r}=E <4-135> (4.7.11) \frac{K}{r}=V(r) <4-136> \nabla V(r)=\vec{i}\frac{\partial V(r)}{\partial x} +\vec{j}\frac{\partial V(r)}{\partial y} +\vec{k}\frac{\partial V(r)}{\partial z} <4-137> \begin{array}{rl} \displaystyle{\frac{\partial V(r)}{\partial x}}} &=\displaystyle{\frac{dV(r)}{dr}}\displaystyle{\frac{\partial r}{\partial x}}}\\ \\ &=\displaystyle{-\frac{K}{r^2}}\displaystyle{-\frac{2x}{2\sqrt{x^2+y^2}}} \\ \\ &=\displaystyle{-\frac{K}{r^2}}\displaystyle{\frac{x}{r}}} \end{array} <4-138> \frac{\partial V(r)}{\partial x}=-\frac{K}{r^2}\frac{x}{r} <4-139> \nabla V(r)= -\left(\vec{i}\frac{x}{r}+\vec{j}\frac{y}{r}\right)\frac{K}{r^2} <4-140> \vec{e}_r=\frac{\vec{r}}{r} <4-141> (4.7.12) \nabla V(r)= -\vec{e}_r \frac{K}{r^2} <4-142> (4.7.13) \vec{F}=-\nabla V(r) <4-143> (4.7.14) v=\pm\frac{2}{m}\sqrt{E-\frac{h^2}{2mr^2}-\frac{K}{r} <4-144> (4.7.15) E\ge\frac{h^2}{2mr^2}+\frac{K}{r} <4-145> (4.7.16) Er^2-Kr-\frac{h^2}{2m}\ge0 <4-146> r=\frac{K}{2E} <4-147> \frac{mK^2+h^2E}{2mE} <4-148> D=K^2-\frac{2h^2E}{m} <4-149> (4.7.17) \left\{ \begin{array}{l} \displaystyle{r_1=\frac{1}{2E}\left[K-\sqrt{K^2-\frac{2h^2E}{m}}\right]} \\ \\ \displaystyle{r_2=\frac{1}{2E}\left[K+\sqrt{K^2-\frac{2h^2E}{m}}\right]} \end{array}\right. <4-150> (4.7.18) \frac{dr}{dt}=\pm\frac{2}{m}\sqrt{E-\frac{h^2}{2mr^2}-\frac{K}{r}} <4-151> (4.7.19) mr^2\omega=h <4-152> (4.7.20) r=\frac{p}{1-e\sin(\theta-\theta_0)},\quad\left\{ \begin{array}{l} p=-\displaystyle{\frac{h^2}{mK}} \\ \\ e=\displaystyle{\sqrt{1+\frac{2Eh^2}{mK^2}}} \end{array}\right. <4-153> (4.8.1) \left\{ \begin{array}{l} \vec{F}_{21}=-\displaystyle{\frac{\vec{r}_2-\vec{r}_1}{\left|\vec{r}_2-\vec{r}_1\right|} \frac{GmM}{\left|\vec{r}_2-\vec{r}_1\right|^2}}\\ \\ \vec{F}_{12}=-\displaystyle{\frac{\vec{r}_1-\vec{r}_2}{\left|\vec{r}_1-\vec{r}_2\right|} \frac{GmM}{\left|\vec{r}_1-\vec{r}_2\right|^2}} \end{array}\right. <4-154> (4.8.2) \left\{ \begin{array}{l} m\displaystyle{\frac{d^2\vec{r}_2}{dt^2}}=\vec{F}_{21} \\ \\ M\displaystyle{\frac{d^2\vec{r}_1}{dt^2}}=\vec{F}_{12} \end{array}\right. <4-155> m\frac{d^2\vec{r}}{dt^2}=-\vec{e}_r\frac{GmM}{r^2} <4-156> (4.8.3) E=\frac{m}{2}\left(\frac{dr}{dt}\right)^2+\frac{h^2}{2mr^2}-\frac{GmM}{r} <4-157> (4.8.4) r=\frac{p}{1+e\cos\theta},\quad\mbox{where}\left\{ \begin{array}{l} p=\displaystyle{\frac{h^2}{Gm^2M}} \\ \\ e=\displaystyle{\sqrt{1-\frac{2|E|h^2}{G^2m^3M^2}}} \end{array}\right. <4-158> (4.8.5) \left\{ \begin{array}{l} \displaystyle{\frac{p}{1-e^2}=\frac{GmM}{2|E|}=a} \\ \\ \displaystyle{\frac{p}{\sqrt{1-e^2} }=\frac{h}{\sqrt{m|E|}}=b} \\ \\ \displaystyle{\frac{ep}{e^2-1}=x_0} \end{array}\right. <4-159> (4.8.6) \frac{(x-x_0)^2}{a^2}+\frac{y^2}{b^2}=1 <4-160> (4.8.7) A=\pi ab <4-161> \frac{dA}{dt}=\frac{1}{2}r^2\frac{d\theta}{dt} <4-162> mr^2\frac{d\theta}{dt}=h <4-163> \frac{dA}{dt}=\frac{h}{2m} <4-164> (4.8.8) T=\frac{A}{(h/2m)}=\frac{2\pi mab}{h} <4-165> (4.8.9) b^2=\frac{2h^2}{Gm^2M}a <4-166> (4.8.10) T^2=\frac{8\pi^2}{GM}a^3 <4-167> (4.9.1) F=G\frac{Mm}{R^2} <4-168> (4.9.2) \left\{ \begin{array}{l} G=6.67\times10^{-11}\left[\mbox{m}^3\mbox{s}^{-2}\mbox{kg}^{-1}\right] \\ \\ M=5.97\times10^{24}\left[\mbox{kg}\right] \\ \\ R=6.38\times10^6\left[\mbox{m}\right] \end{array}\right. <4-169> (4.9.3) g=G\frac{M}{R^2}=9.8\left[\mbox{m}/\mbox{s}^{2}\right] <4-170> (4.9.4) F=mg <4-171> \pi\times\frac{5}{180}\sim0.0873 <4-172> \pi\times\frac{10}{180}\sim0.1745 <4-173> \sin5^\circ\sim0.0872 <4-174> \sin10^\circ\sim0.1736 <5-1> (5.1.1) \left\{ \begin{array}{l} m_1\displaystyle{\frac{d^2\vec{r}_1(t)}{dt^2}} =\vec{F}_{21} (|\vec{r}_2-\vec{r}_1|) \\ \\ m_2\displaystyle{\frac{d^2\vec{r}_2(t)}{dt^2}} =\vec{F}_{12} (|\vec{r}_1-\vec{r}_2|) \end{array}\right. <5-2> |\vec{r}_1-\vec{r}_2 |=|\vec{r}_2-\vec{r}_1|\equiv r <5-3> (5.1.2) \vec{F}_{21} (r) =-\vec{F}_{12} (r) <5-4> (5.1.3) \left\{ \begin{array}{l} \vec{p}_1(t)=m_1\displaystyle{\frac{d\vec{r}_1(t)}{dt}} \equiv m_1\vec{v}_1(t) \\ \\ \vec{p}_2(t)=m_2\displaystyle{\frac{d\vec{r}_2(t)}{dt}} \equiv m_2\vec{v}_2(t) \end{array}\right. <5-5> (5.1.4) \left\{ \begin{array}{l} \displaystyle{\frac{d\vec{p}_1(t)}{dt}}= \vec{F}_{21}(r) \\ \\ \displaystyle{\frac{d\vec{p}_2(t)}{dt}}= \vec{F}_{12}(r) \end{array}\right. <5-6> (5.1.5) \frac{d\vec{p}_1(t)}{dt}+\frac{d\vec{p}_2(t)}{dt}=0 <5-7> (5.1.6) \vec{P}(t) =\vec{p}_1(t) +\vec{p}_2(t) <5-8> (5.1.7) \frac{d\vec{P}(t)}{dt}=0 <5-9> (5.1.8) \vec{P}(t) =\vec{P}_0 <5-10> (5.1.9) \vec{R}(t)=\frac{m_1\vec{r}_1(t)+ m_2\vec{r}_2(t)}{m_1+m_2} <5-11> \begin{array}{rl} \displaystyle{\frac{d\vec{R}}{dt}} &=\displaystyle{\frac{m_1}{m_1+m_2}\frac{d\vec{r}_1}{dt} +\frac{m_2}{m_1+m_2}\frac{d\vec{r}_2}{dt}} \\ \\ &=\displaystyle{\frac{1}{m_1+m_2}}\left(\vec{p}_1+\vec{p}_2\right) \\ \\ &=\displaystyle{\frac{1}{ m_1+m_2}}\vec{P} \end{array} <5-12> (5.1.10) \vec{P}(t)=(m_1+m_2)\frac{d\vec{R}(t)}{dt} <5-13> (5.1.11) \begin{array}{l} \displaystyle{m_1\frac{d^2\vec{r}_1(t)}{dt^2} =\vec{F}_{21}(r_{21})+\vec{F}_{31}(r_{31})+\cdots+\vec{F}_{N1}(r_{N1})} \\ \\ \displaystyle{m_2\frac{d^2\vec{r}_2(t)}{dt^2} =\vec{F}_{12}(r_{12})+\vec{F}_{32}(r_{32})+\cdots+\vec{F}_{N2}(r_{N2})} \\ \\ \cdots\cdots\cdots\cdots\cdots \\ \\ \displaystyle{m_N\frac{d^2\vec{r}_N(t)}{dt^2} =\vec{F}_{1N}(r_{1N})+\vec{F}_{2N}(r_{2N})+\cdots+\vec{F}_{N-1,N}(r_{N-1,N})} \end{array} <5-14> (5.1.12) \left\{ \begin{array}{l} \vec{p}_1(t) =m_1\displaystyle{\frac{d\vec{r}_{1}(t)}{dt}}\equiv m_1\vec{v}_1(t) \\ \\ \vec{p}_2(t) =m_2\displaystyle{\frac{d\vec{r}_{2}(t) }{dt}}\equiv m_2\vec{v}_2(t) \\ \\ \cdots\cdots\cdots\cdots\cdots \\ \\ \vec{p}_N(t) =m_N\displaystyle{\frac{d\vec{r}_{N}(t) }{dt}}\equiv m_N\vec{v}_N(t) \end{array}\right. <5-15> (5.1.13) \left\{\begin{array}{l} \displaystyle{\frac{d\vec{p}_1(t)}{dt}}=\vec{F}_{21}(r_{21})+\vec{F}_{31}(r_{31})+\cdots +\vec{F}_{N1}(r_{N1}) \\ \\ \displaystyle{\frac{d\vec{p}_2(t)}{dt}}=\vec{F}_{12}(r_{12})+\vec{F}_{32}(r_{32})+\cdots +\vec{F}_{N2}(r_{N2}) \\ \\ \cdots\cdots\cdots\cdots\cdots \\ \\ \displaystyle{\frac{d\vec{p_N}(t)}{dt}}=\vec{F}_{1N}(r_{1N})+\vec{F}_{2N}(r_{2N})+\cdots +\vec{F}_{N-1,N}(r_{N-1,N}) \end{array}\right. <5-16> (5.1.14) \vec{F}_{ij}(r_{ij})=-\vec{F}_{ji}(r_{ji})=-\vec{F}_{ji}(r_{ij}) <5-17> (5.1.15) \frac{d\vec{P}(t)}{dt}=0 <5-18> (5.1.16) \vec{P}(t)=\vec{p}_1(t)+ \vec{p}_2(t)+\cdots+\vec{p}_N(t) <5-19> (5.1.17) \vec{R}(t)=\frac{m_1\vec{r}_1(t)+m_2\vec{r}_2(t)+\cdots+m_N\vec{r}_N(t)} {m_1+ m_2+\cdots+ m_N} <5-20> (5.1.18) \vec{P}(t)= (m_1+ m_2+\cdots+ m_N)\frac{d\vec{R}(t) }{dt} <5-21> (5.2.1) \vec{r}_1(t)-\vec{r}_2(t)\equiv\vec{r}(t) <5-22> (5.2.2) \left\{ \begin{array}{l} m_1\displaystyle{\frac{d^2\vec{r}_{1}(t)}{dt}}=\vec{F}_{21}(r) \\ \\ m_2\displaystyle{\frac{d^2\vec{r}_{2}(t)}{dt}}=\vec{F}_{12}(r) \end{array}\right. <5-23> \vec{F}_{21}(r) \equiv \vec{F} (r) <5-24> m_1m_2\frac{d^2\vec{r}(t)}{dt^2}}=( m_1+m_2)\vec{F} (r) <5-25> (5.2.3) \mu=\frac{m_1m_2}{m_1+m_2} <5-26> (5.2.4) \mu\frac{d^2\vec{r}(t)}{dt^2}}=\vec{F} (r) <5-27> (5.2.5) M\frac{d^2\vec{R}}{dt^2}}=0 <5-28> \left\{ \begin{array}{l} m_1\displaystyle{\frac{d^2\vec{r}_{1}(t)}{dt^2}} =\vec{F}_{21}(\left|\vec{r}_1-\vec{r}_2\right|) \\ \\ m_2\displaystyle{\frac{d^2\vec{r}_{2}(t)}{dt^2}} =\vec{F}_{12}(\left|\vec{r}_2-\vec{r}_1\right|) \end{array}\right. <5-29> \left\{ \begin{array}{l} M\displaystyle{\frac{d^2\vec{R}(t)}{dt^2}}=0 \\ \\ \mu\displaystyle{\frac{d^2\vec{r}(t)}{dt^2}}=\vec{F} (r) \end{array}\right. <5-30> (5.2.6) \left\{ \begin{array}{l} \vec{R}(t)=\displaystyle{\frac{m_1\vec{r}_1(t)+ m_2\vec{r}_2(t)}{ m_1+ m_2}} \\ \\ \vec{r}(t)=\vec{r}_1(t)-\vec{r}_2 (t) \end{array}\right. <5-31> \vec{F}(r)\equiv\vec{F}_{21} (|\vec{r}_1-\vec{r}_2|) =- \vec{F}_{12} (|\vec{r}_2-\vec{r}_1|) <5-32> (5.2.7) \left\{\begin{array}{l} \displaystyle{\vec{r}_1=\vec{R}+\frac{m_2}{M}\vec{r}} \\ \\ \displaystyle{\vec{r}_2=\vec{R}-\frac{m_1}{M}\vec{r}} \end{array}\right. <5-33> (5.2.8) \begin{array}{rl} \displaystyle{\frac{m_1\vec{v}_1^2}{2}}+\displaystyle{\frac{m_2\vec{v}_2^2}{2}} &=\displaystyle{\frac{m_1}{2}\left(\frac{d\vec{R}}{dt} +\frac{\vec{m_2}}{M}\frac{d\vec{r}}{dt}\right)^2} +\frac{m_2}{2}\left(\frac{d\vec{R}}{dt} -\frac{\vec{m_1}}{M}\frac{d\vec{r}}{dt}\right)^2} \\ \\ &=\displaystyle{\frac{M}{2}\left(\frac{d\vec{R}}{dt}\right)^2 +\frac{\mu}{2}\left(\frac{d\vec{r}}{dt}\right)^2} \end{array} <5-34> (5.3.1) \begin{array}{rl} \displaystyle{m\frac{d^2x_A}{dt^2}} &=F_A+f_A \\ \\ &=-kx_A+k(x_B-x_A) \\ \\ &=-2kx_A+kx_B \end{array} <5-35> (5.3.2) \begin{array}{rl} \displaystyle{m\frac{d^2x_B}{dt^2}} &=F_B+f_B \\ \\ &=-kx_B-k(x_B-x_A) \\ \\ &=kx_A-2kx_B \end{array} <5-36> \frac{(\ell+x_A)+(2\ell+x_B)}{2}=\frac{x_A+x_B}{2}+\frac{3}{2}\ell <5-37> (2\ell+x_B)-(\ell+x_A)=x_B-x_A+\ell <5-38> (5.3.3) \left\{\begin{array}{rl} X&=\displaystyle{\frac{(x_A+\ell)+(x_B+2\ell)}{2}}-\frac{3\ell}{2}\\ \\ &=\displaystyle{\frac{x_A+x_B}{2}} \\ \\ x&=\displaystyle{(x_B-x_A+\ell)-\ell} \\ \\ &=\displaystyle{x_B-x_A} \end{array}\right. <5-39> (5.3.4) M\frac{d^2X}{dt^2}=-2kX <5-40> (5.3.5) \mu\frac{d^2x}{dt^2}=-\frac{3k}{2}x <5-41> (5.3.6) X(t)=A\sin(\Omega t+B) <5-42> \Omega=\sqrt\frac{2k}{M}} <5-43> (5.3.7) x(t)=a\sin(\omega t+b) <5-44> \omega=\sqrt{\frac{3k}{2\mu}} <5-45> (5.3.8) \left\{\begin{array}{l} x_A=X-\displaystyle{\frac{x}{2}}\\ \\ x_B=X+\displaystyle{\frac{x}{2}} \end{array}\right. <5-46> (5.3.9) \left\{\begin{array}{l} \displaystyle{x_A=A\sin(\Omega t+B)-\frac{a}{2}\sin(\omega t+b)}\\ \\ \displaystyle{x_B= A\sin(\Omega t+B)+\frac{a}{2}\sin(\omega t+b)} \end{array}\right. <5-47> (5.4.1) \left\{\begin{array}{l} \vec{r}=\vec{r}_1-\vec{r}_2\\ \\ \displaystyle{\vec{R}= \frac{m_1\vec{r}_1+m_2\vec{r}_1}{m_1+m_2}} \end{array}\right. <5-48> (5.4.2) \left\{\begin{array}{l} \vec{v}=\vec{v}_1-\vec{v}_2\\ \\ \displaystyle{\vec{V}= \frac{m_1\vec{v}_1+m_2\vec{v}_1}{M}} \end{array}\right. <5-49> (5.4.3) \left\{\begin{array}{l} \displaystyle{\vec{v}_1=\vec{V}+\frac{m_2}{M}\vec{v}}\\ \\ \displaystyle{\vec{v}_2=\vec{V}-\frac{m_1}{M}\vec{v}} \end{array}\right. <5-50> (5.4.4) \begin{array}{rl} K&=\displaystyle{\frac{m_1}{2}\vec{v}_1^2+\frac{m_2}{2}\vec{v}_2^2} \\ \\ &=\displaystyle{\frac{m_1}{2}\left(\vec{V}+\frac{m_2\vec{v}}{M}\right)^2+ \frac{m_2}{2}\left(\vec{V}-\frac{m_1\vec{v}}{M}\right)^2} \\ \\ &=\displaystyle{\frac{M}{2}\vec{V}^2+\frac{m_1m_2}{2(m_1+m_2)}\vec{v}^2} \\ \\ &=\displaystyle{\frac{M}{2}\vec{V}^2+\frac{\mu}{2}\vec{v}^2} \end{array} <5-51> (5.4.5) \left\{\begin{array}{l} \vec{v}^\prime=\vec{v}_1^\prime -\vec{v}_2 ^\prime \\ \\ \displaystyle{\vec{V}^\prime =\frac{m_1\vec{v}_1^\prime +m_2\vec{v}_1^\prime }{M}} \end{array}\right. <5-52> (5.4.6) \left\{\begin{array}{l} \displaystyle{\vec{v}_1^\prime=\vec{V}^\prime+\frac{m_2}{M}\vec{v}^\prime} \\ \\ \displaystyle{\vec{v}_2^\prime=\vec{V}^\prime-\frac{m_1}{M}\vec{v}^\prime} \end{array}\right. <5-53> (5.4.7) \begin{array}{rl} K^\prime &=\displaystyle{\frac{m_1}{2}}\vec{v}_1^{\prime 2}+\frac{m_2}{2} \vec{v}_2^{\prime2}}\\ \\ &=\displaystyle{\frac{M}{2}}\vec{V}^{\prime2}+\frac{\mu}{2}\vec{v}^{\prime2}} \end{array} <5-54> \vec{P}=m_1\vec{v}_1+m_2\vec{v}_2=M\vec{V} <5-55> \vec{P}^\prime =m_1\vec{v}_1^\prime +m_2\vec{v}_2^\prime =M\vec{V}^\prime <5-56> (5.4.8) K^\prime=\frac{M}{2}}\vec{V}^{2}+\frac{\mu}{2}}\vec{v}^{\prime2} <5-57> (5.4.9) K-K^\prime=\frac{\mu}{2}}\vec{v}^{2} \left[1-\frac{\left(\vec{v}_1^\prime-\vec{v}_2^\prime\right)^2}{ \left(\vec{v}_1 -\vec{v}_2 \right)^2}\right] <5-58> (5.4.10) e=\frac{|\vec{v}_1^\prime-\vec{v}_2^\prime|}{|\vec{v}_1 -\vec{v}_2|} <5-59> (5.4.11) K-K^\prime=\frac{\mu}{2}(1-e^2) <5-60> (5.4.12) 0\le e\le 1 <5-61> (5.4.13) e=-\frac{\vec{v}_1^\prime-\vec{v}_2^\prime}{\vec{v}_1 -\vec{v}_2} <5-62> (5.4.14) \begin{array}{l} mv_1+mv_2=mv_1^\prime+ mv_2^\prime \\ \\ \quad\Rightarrow\quad v_1+v_2= v_1^\prime+ v_2^\prime \end{array} <5-63> (5.4.15) \begin{array}{l} \displaystyle{\frac{1}{2}mv_1^2+\frac{1}{2}mv_2^2=\frac{1}{2}mv_1^{\prime2}+ \frac{1}{2}mv_2^{\prime2}} \\ \\ \quad\Rightarrow\quad v_1^2+v_2^2= v_1^{\prime2}+ v_2^{\prime2} \end{array} <5-64> (5.4.16) v_1^\prime=v_2^\prime <5-65> (v_1+v_2)^2-2v_1v_2=(v_1^\prime+v_2^\prime)^2-2 v_1^\prime v_2^\prime <5-66> (5.4.17) v_1v_2=v_1^\prime v_2^\prime <5-67> (v_1-v_2)^2=(v_1^\prime-v_2^\prime)^2 <5-68> (5.4.18) \left\{ \begin{array} {ll} (1) & v_1-v_2=v_1^\prime-v_2^\prime \\ \\ (2) & v_1-v_2=-(v_1^\prime-v_2^\prime) \end{array}\right. <5-69> (5.4.19) \begin{array} {rl} mv_1+mv_2&=mv_1^\prime+mv_2^\prime \\ \\ &=2mv_1^\prime \\ \\ &=2mv_2^\prime \end{array} <5-70> (5.4.20) v_1^\prime=v_2^\prime=\frac{v_1+v_2}{2} <5-71> |\vec{a}|=\sqrt{\vec{a}\cdot\vec{a}} <5-72> |\vec{a}|=\sqrt{a_x^2+ a_y^2+ a_z^2} <6-1> (6.1.1) \begin{array}{l} |\vec{r}_i-\vec{r}_j|=a \\ \\ \quad(i=1,2,\ldots,n; j=1,2,\ldots,n;i\ne j) \end{array} <6-2> (6.1.2) \vec{r}=\frac{m(\vec{r}_1+\vec{r}_2+\cdots+\vec{r}_n)}{M} <6-3> (6.2.1) \left\{\begin{array}{l} x_k=a_k\cos\theta_k \\ \\ y_k=a_k\sin\theta_k \end{array}\right. <6-4> (6.2.2) \vec{v}_k=\vec{i}v_{kx}+\vec{j}v_{ky}+\vec{k}v_{kz} <6-5> (6.2.3) \left\{\begin{array}{l} \displaystyle{v_{kx}=\frac{dx_k}{dt}=-a_k\omega\sin\theta_k } \\ \\ \displaystyle{v_{ky}=\frac{dy_k}{dt}=a_k\omega\cos\theta_k} \\ \\ \displaystyle{v_{kz}=\frac{dz_k}{dt}=0} \end{array}\right. <6-6> (6.2.4) v_k=\sqrt{v_{kx}^2+ v_{ky}^2+ v_{kz}^2}=a_k\omega <6-7> \vec{l}_k=\vec{r}_k\times\vec{p}_k <6-8> (6.2.5) \left\{\begin{array}{ll} \vec{l}_k&=m\left[\vec{i}(y_kv_{kz}-z_kv_{ky})+\vec{j}(z_kv_{kx}-x_kv_{kz})\right.\\ \\ &\quad+\left.\vec{k}(x_kv_{ky}-y_kv_{kx})\right]\\ \\ &=\vec{k}(ma_k^2\omega) \end{array}\right. <6-9> (6.2.6) \vec{l}=\sum_{k=1}^{n}\vec{l}_k =\vec{k}\left(\sum_{k=1}^{n}ma_k^2\right)\omega <6-10> (6.2.7) I=\sum_{k=1}^{n}ma_k^2 <6-11> (6.2.8) \vec{l}=\vec{k}I\omega <6-12> (6.2.9) \vec{l}=\sum_{k=1}^{n}\vec{l}_k=I\vec{\omega} <6-13> \vec{l}_k=\vec{r}_k\times\vec{p}_k <6-14> \frac{d\vec{l}_k}{dt}=\frac{d\vec{r}_k}{dt}\times\vec{p}_k +\vec{r}_k\times\frac{d\vec{p}_k}{dt} <6-15> \frac{d\vec{r}_k}{dt}= \vec{v}_k=\frac{1}{m}\vec{p}_k <6-16> (6.2.10) \frac{d\vec{l }_k}{dt}=\vec{r}_k\times\vec{F}\equiv\vec{N} <6-17> I\frac{d\vec{\omega}}{dt}=\frac{d\vec{l}_k}{dt} <6-18> (6.2.11) I\frac{d\vec{\omega}}{dt}=\vec{N} <6-19> (6.2.12) \vec{\omega}=\mbox{a constant vector} <6-20> \left(\vec{N}_1=\vec{r}_1\times\vec{F}_1, \vec{N}_2=\vec{r}_2\times\vec{F}_2,\ldots, \vec{N}_n=\vec{r}_n\times\vec{F}_n\right) <6-21> (6.2.13) I\frac{d\vec{\omega}}{dt}=\sum_{k=1}^{n} \left(\vec{r}_k\times\vec{F}_k\right) =\sum_{k=1}^{n} \vec{N}_k <6-22> (6.3.1) I=I_G+Md^2 <6-23> (6.3.2) I_z=I_x+I_y <6-24> (6.4.1) I_1= M\ell^2 <6-25> (6.4.2) \begin{array}{rl} I_2&=\displaystyle{M\left(\frac{\ell}{2}\right)^2+ M\left(\frac{\ell}{2}\right)^2} \\ \\ &=\displaystyle{\frac{M\ell^2}{2} \end{array} <6-26> (6.4.3) \begin{array}{rl} I_2&=\displaystyle{I_1-2M\left(\frac{\ell}{2}\right)^2} \\ \\ &= \displaystyle{M\ell^2-2M\left(\frac{\ell}{2}\right)^2} \\ \\ &=\displaystyle{ \frac{M\ell^2}{2}} \end{array} <6-27> (6.4.4) \begin{array}{rl} I_1&=\displaystyle{m\left[\left(\frac{\ell}{2N}\right)^2+\left(2\frac{\ell}{2N}\right)^2 +\ldots+\left(2N\frac{\ell}{2N}\right)^2\right]} \\ \\ &= \displaystyle{ m\left(\frac{\ell}{2N}\right)^2\left[1^2+2^2+\ldots+(2N)^2\right]} \\ \\ &=\displaystyle{ \frac{m\ell^2}{4N^2}\times\frac{(2N)(2N+1)(4N+1)}{6}} \\ \\ &=\displaystyle{ \frac{M\ell^2}{8N^3}\times\frac{(2N)(2N+1)(4N+1)}{6}} \quad(\because 2m=M) \end{array} <6-28> (6.4.5) \sum_{k=1}^{N} k^2=\frac{N(N+1)(2N+1)}{6} <6-29> I_1=\frac{M\ell^2}{8}\frac{2\{2+(1/N)\}\{4+(1/N)\}}{6} <6-30> I_1=\frac{1}{3}M\ell^2 <6-31> (6.4.7) \begin{array}{rl} I_2&=\displaystyle{2\times m\left[\left(\frac{\ell}{2N}\right)^2 +\left(2\frac{\ell}{2N}\right)^2+\cdots+\left(N\frac{\ell}{2N}\right)^2\right]} \\ \\ &=\displaystyle{ 2\times\frac{M\ell^2}{8N^3}\times\frac{(2N)(2N+1)(4N+1)}{6}} \quad(\because 2m=M) \end{array} <6-32> (6.4.8) I_2=\frac{1}{12}M\ell^2 <6-33> (6.4.9) I_1=I_2+M\left(\frac{ \ell}{2}\right)^2 <6-34> (6.4.10) I_2=I_1-M\left(\frac{ \ell}{2}\right)^2=\frac{1}{12}M\ell^2 <6-35> \frac{1}{3}M\ell^2 <6-36> \frac{1}{12}M\ell^2 <6-37> M\ell^2 <6-38> \frac{1}{2}M\ell^2 <6-39> \frac{1}{8}Mb^2 <6-40> \frac{M}{8}(a^2+b^2) <6-41> \frac{M}{4}a^2 <6-42> \frac{M}{2}a^2 <6-43> \frac{2M}{5}a^2 <6-44> \frac{M}{4}a^2+\frac{M}{3}h^2 <6-45> \frac{M}{2}a^2