# 自然科学/功 动能

## 功

### 机械功

1. ${\displaystyle 0\leqslant \theta <{\frac {\pi }{2}}}$ ${\displaystyle W>0}$ ，称${\displaystyle {\boldsymbol {F}}}$ 做正功
2. ${\displaystyle \theta ={\frac {\pi }{2}}}$ ${\displaystyle W=0}$ ，称${\displaystyle {\boldsymbol {F}}}$ 不做功
3. ${\displaystyle {\frac {\pi }{2}}<\theta \leqslant \pi }$ ${\displaystyle W<0}$ ，称${\displaystyle {\boldsymbol {F}}}$ 做负功

### 机械功的物理意义

{\displaystyle {\begin{aligned}W&={\boldsymbol {G}}\cdot {\boldsymbol {s}}\\&=(0,mg)\cdot (v_{0}t,{\frac {1}{2}}gt^{2})\\&={\frac {1}{2}}mg^{2}t^{2}\end{aligned}}}

### 合力与分力的功

${\displaystyle {\boldsymbol {F}}={\boldsymbol {F_{1}}}+{\boldsymbol {F_{2}}}+\cdots +{\boldsymbol {F_{n}}}}$ 根据功的定义，任何一个分力做功

${\displaystyle W_{i}={\boldsymbol {F_{i}}}\cdot {\boldsymbol {s}}}$ 因此有

{\displaystyle {\begin{aligned}W&={\boldsymbol {F}}\cdot {\boldsymbol {s}}\\&=({\boldsymbol {F_{1}}}+{\boldsymbol {F_{2}}}+\cdots +{\boldsymbol {F_{n}}})\cdot {\boldsymbol {s}}\\&=({\boldsymbol {F_{1}}}\cdot {\boldsymbol {s}})+({\boldsymbol {F_{2}}}\cdot {\boldsymbol {s}})+\cdots +({\boldsymbol {F_{n}}}\cdot {\boldsymbol {s}})\\&=W_{1}+W_{2}+\cdots +W_{n}\end{aligned}}} 因此合力做功等于每一个分力做功之和。

## 动能

${\displaystyle E_{k}={\frac {1}{2}}mv^{2}}$ 其中${\displaystyle v}$ 为物体相对于参考系的速度大小。同物体的动能侧面反映了物体所具有的维持其惯性的能量。

## 动能定理

${\displaystyle {\boldsymbol {v_{0}}}=(v_{x},v_{y})}$

{\displaystyle {\begin{aligned}E_{k}&={\frac {1}{2}}m|{\boldsymbol {v_{0}}}|^{2}\\&={\frac {1}{2}}m(v_{x}^{2}+v_{y}^{2})\\\end{aligned}}}

${\displaystyle {\boldsymbol {F}}=(F_{x},F_{y})}$

${\displaystyle {\boldsymbol {s}}={\boldsymbol {v_{0}}}t+{\frac {\boldsymbol {F}}{2m}}t^{2}}$ 因此合力做功{\displaystyle {\begin{aligned}W&={\boldsymbol {F}}\cdot {\boldsymbol {s}}\\&={\boldsymbol {v_{0}F}}t+{\frac {({\boldsymbol {F}}t)^{2}}{2m}}\\&=v_{x}F_{x}t+v_{y}F_{y}t+{\frac {F_{x}^{2}t^{2}}{2m}}+{\frac {F_{y}^{2}t^{2}}{2m}}\\\end{aligned}}} 现在让我们来看看${\displaystyle t_{1}}$ 时刻物体的动能，我们先计算出此时的速度 {\displaystyle {\begin{aligned}{\boldsymbol {v_{1}}}&={\boldsymbol {v_{0}}}+{\frac {{\boldsymbol {F}}t}{m}}\\&=(v_{x}+{\frac {F_{x}t}{m}},v_{y}+{\frac {F_{y}t}{m}})\end{aligned}}} 因此，此时物体的动能为{\displaystyle {\begin{aligned}E_{k}'&={\frac {1}{2}}m|{\boldsymbol {v_{1}}}|^{2}\\&={\frac {1}{2}}mv_{x}^{2}+{\frac {1}{2}}mv_{y}^{2}+v_{x}F_{x}t+v_{y}F_{y}t+{\frac {F_{x}^{2}t^{2}}{2m}}+{\frac {F_{y}^{2}t^{2}}{2m}}\end{aligned}}} 因此有

${\displaystyle W=E_{k}'-E_{k}}$ 对于不断的合力做功，我们可以将变力看做很多微小位移段上的恒力，于是，综合起来我们可以得到，动能定理：物体所受到合力所做的功等于物体动能变化量。

## *引力势能

${\displaystyle E_{p}=-{\frac {Gm_{A}m_{B}}{r}}}$ 其中${\displaystyle r}$ 为两个物体间的距离，${\displaystyle m_{A}}$ ${\displaystyle m_{B}}$ 分别为两个物体的质量，${\displaystyle G}$ 为万有引力常量。可见，将两个物体中的任何一个物体作为参照，另一个物体的引力势能表达式都是一样的。

{\displaystyle {\begin{aligned}E_{p}&=W\\&=\textstyle \int _{0}^{r}-{\frac {Gm_{A}m_{b}}{x^{2}}}dx\\&=-{\frac {Gm_{A}m_{b}}{r}}\end{aligned}}}