# 向心力

## 公式(代數證法)

1. 一物體要做勻速圓周運動所需要的向心力大小為:

${\displaystyle F=m\omega ^{2}r}$

2. 欲知向心力與線速度大小的關係,可以將${\displaystyle \omega ={\frac {v}{r}}}$代入${\displaystyle F=m\omega ^{2}r}$,也就是物體的線速度與其角速度的關係:

${\displaystyle F=m\omega ^{2}r}$
${\displaystyle F=m{\frac {v^{2}}{r}}}$

${\displaystyle \mathbf {F_{c}} =-{\frac {mv^{2}}{r}}{\hat {\mathbf {r} }}=-{\frac {mv^{2}}{r}}{\frac {\mathbf {r} }{r}}=-m\omega ^{2}\mathbf {r} =m{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times {\boldsymbol {r}})}$
• 註：${\displaystyle {\hat {\mathbf {r} }}}$表示${\displaystyle \mathbf {r} }$方向的單位向量

3. 因此由上方的公式表述，從牛頓定律的帶入可得知，

## 向心加速度之推導(畢氏三角證法)

${\displaystyle \mathbf {R} }$ = ${\displaystyle {\boldsymbol {r+d}}}$(半徑加上物體瞬間之掉落距離) 所以 ${\displaystyle \mathbf {d} }$ = ${\displaystyle {\boldsymbol {R-r}}}$ 由於 ${\displaystyle \mathbf {d} }$ = ${\displaystyle {\frac {1}{2}}{\boldsymbol {a\Delta t^{2}}}}$; 則 ${\displaystyle \mathbf {a} }$= ${\displaystyle \left({\frac {2d}{\Delta t^{2}}}\right)}$

${\displaystyle a=\lim _{\Delta t\to 0}{\frac {2d}{\Delta t^{2}}}}$

${\displaystyle a=\lim _{\Delta t\to 0}{\frac {2({\sqrt {r^{2}+(v\Delta t)^{2}}}-r)}{\Delta t^{2}}}}$

${\displaystyle a={\frac {2(v^{2})}{{\sqrt {r^{2}}}+r}}}$

${\displaystyle a={\frac {2(v^{2})}{2r}}}$