8-43 할리데이 11판 솔루션 일반물리학

$$\put \begin{cases} U : \text{Potential Energy}\\ \end{cases}$$ $$\begin{cases} F(x)&=G\cfrac{m_1m_2}{x^2}\\ U(\infin)&=0\\ \end{cases}$$ $$\begin{cases} x&\gt0\\ x_1&=x_1\\ x_2&=x_1+d \end{cases}$$ $$\ab{a}$$ $$\begin{aligned} U(x)&=U(\infin)+\int_{\infin}^xF(x)\dd x\\ &=0+\int_{\infin}^x G\frac{m_1m_2}{x^2}\dd x\\ &=Gm_1m_2\int_{\infin}^x \frac{1}{x^2}\dd x\\ &=-Gm_1m_2\[\frac{1}{x}\]_{\infin}^x\\ &=-Gm_1m_2\(\frac{1}{x}-\frac{1}{\infin}\)\\ &=-Gm_1m_2\(\frac{1}{x}\)\\ &=-G\frac{m_1m_2}{x}\\ \end{aligned}$$ $$\ab{b}$$ $$\begin{aligned} W_{x_1\rarr x_2}&=\Delta U_{x_1\rarr x_2}\\ &=U(x_2)-U(x_1)\\ &=\(-G\frac{m_1m_2}{x_2}\)-\(-G\frac{m_1m_2}{x_1}\)\\ &=Gm_1m_2\(\frac{1}{x_1}-\frac{1}{x_2}\)\\ &=Gm_1m_2\(\frac{1}{x_1}-\frac{1}{x_1+d}\)\\ &=G\frac{m_1m_2}{x_1}\cdot\frac{d}{d+{x_1}}\\ &=U(x_1)\cdot\frac{d}{d+{x_1}} \end{aligned}$$