███████████ █████

███████████ █████ (from █████ ███████ '██████' and ██████ 'to ████') is the █████ that █████ a ████ ██████ a ██████ ████. The █████████ of the ███████████ █████ is ██████ ██████████ to the ██████ of the ████ and ███████ the █████ █████ of the █████████████ ██████ of █████████ of the ████. █████ ██████ ██████ the ████, ██████████ it as "a █████ by which ██████ are █████ or ████████, or in ███ ███ ████, ███████ a █████ as to a ██████". In █████████ █████████, ███████ ████████ the ███████████ █████ ███████ ████████████ ██████.

███ ██████ ███████ █████████ ███████████ █████ is the ████ in which a ████ █████ with ███████ █████ along a ████████ ████. The ███████████ █████ is ████████ at █████ ██████ to the ██████ and ████ along the ██████ ███████ the ██████ of the ████████ ████. The ████████████ ███████████ was ███████ in ████ by the █████ █████████ ██████████ ███████.

███████

From the ██████████ of ██████ ██████ it is █████ that an ██████ ██████ at ██████████ █████ █ along a ████ with ██████ of █████████ █ ███████████ toward the ██████ of █████████ at a ████ ${\textbf {a}}_{c}=\lim _{\Delta t\to 0}{\frac {\Delta {\textbf {v}}}{\Delta t}},\quad a_{c}={\frac {v^{2}}{r}}$ ████, $a_{c}$ is the ███████████ ████████████ and $\Delta {\textbf {v}}$ is the ██████████ between the ████████ ███████ at $t+\Delta {t}$ and $█$ .

By ██████'s ██████ ███, the █████ of ████████████ is a ███ █████ ██████ on the ██████, which is ████████████ to ███ ████ █ and ███ ████████████. The █████, ███████ ████████ to as a ███████████ █████, has a █████████ $F_{c}=ma_{c}=m{\frac {v^{2}}{r}}$ and is, like ███████████ ████████████, ████████ toward the ██████ of █████████ of the ██████'s ██████████.

██████████

The ███████████ ████████████ can be ████████ from the ███████ of the ████████ ███████ at ███ █████████. In the ████ of ███████ ████████ ██████ the ██████████ have ████████ █████████. Because ████ ███ is █████████████ to ███ ██████████ ████████ ██████, ██████ ██████ ███████████ ███████ ███ ███████ █████████ █████████ with █████████ ██████ – ███ ██████████ a ████ of $\Delta {\textbf {v}}$ and a ███ ██████ of $█$ , and the █████ a ████ of $\Delta {\textbf {r}}$ (████████ ██████ ██████████) and a ███ ██████ of $█$ ${\frac {|\Delta {\textbf {v}}|}{v}}={\frac {|\Delta {\textbf {r}}|}{r}}$ $|\Delta {\textbf {v}}|={\frac {v}{r}}|\Delta {\textbf {r}}|$ █████████, $|\Delta {\textbf {v}}|$ can be ███████████ with ${\frac {v}{r}}|\Delta {\textbf {r}}|$ : $a_{c}=\lim _{\Delta t\to 0}{\frac {|\Delta {\textbf {v}}|}{\Delta t}}={\frac {v}{r}}\lim _{\Delta t\to 0}{\frac {|\Delta {\textbf {r}}|}{\Delta t}}={\frac {v^{2}}{r}}$ The █████████ of the █████ is toward the ██████ of the ██████ in which the ██████ is ██████, or the ██████████ ██████ (the ██████ that ████ ████ the █████ ████ of the ██████, if the ████ is not ████████). The █████ in the ███████ is ███████, ██ █████ the █████ █████ ████ █████ the █████, at a █████ ██████.

This █████ is ████ █████████ ███████ in █████ of the ███████ ████████ █ of the ██████ about the ██████ of the ██████, ███████ to the ██████████ ████████ by the ███████ $v=\omega r$ ██ that $F_{c}=mr\omega ^{2}\,.$

█████████ using the ███████ ██████ █ for ███ ██████████ of the ██████, $\omega ={\frac {2\pi }{T}}$ the ████████ ███████ $F_{c}=mr\left({\frac {2\pi }{T}}\right)^{2}.$

In ████████ ████████████, ████████ can be ████ ████ (█████ to the █████ of █████ in ██████) ██ the ████ ████ ████ ███ ██████ ███████ ███████ (████████████ ████) ███████ █████████ ███████ █████ for the ████ ███████████ ████████████, ██ the ████████ ███████: $F_{c}={\frac {\gamma mv^{2}}{r}}$ █████ $\gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$ is the ███████ ██████.

████ the ███████████ █████ is █████ by: $F_{c}=\gamma mv\omega$ which is the ████ of ██████ of ████████████ ████████ $\gamma mv$ .

███████

In the ████ of an ██████ that is ████████ around on the ███ of a ████ in a ██████████ █████, the ███████████ █████ on the ██████ is ████████ by the ███████ of the ████. The ████ ███████ is an ███████ █████████ a '████' █████. The ███████████ █████ can ████ be ████████ as a '████' █████, ████ as in the ████ █████ the ██████ ████████ of a ████ ████████ the ███████████ █████ for a ████ of █████ or a █████ █████.

██████'s ████ of a ███████████ █████ ███████████ to ████ is ████████ ████████ to as a ███████ █████. ████ a █████████ is in █████ around a ██████, ███████ is ██████████ to be a ███████████ █████ ████ ██████ in the ████ of █████████ ██████, the █████████████ █████ is ████████ ███████ the █████, and not ███████ the █████████████ ██████ of █████████.

███████ ███████ of ███████████ █████ ██████ in the █████ that is ██████ out ████ a ███████ ████████ █████ in a ███████ ████████ █████ in the ███████ of █████ ████████ ██████. In this ████, the ████████ █████ is the ███████████ █████ that ████ ███████ the █████ ████.

████████ of ███████ █████

Below are █████ ████████ of ██████████ ██████████, with ███████████ of the ████████ █████████ ████████ and ████████████.

███████ ████████ ██████

███████ ████████ ██████ ██████ to the ████ of ████████ ████ of ████████. ████ are ███ ██████████ to ██████████ this ████.

████████ ██████████

In ███ ██████████, the ████████ ██████ ${\textbf {r}}$ , which has █████████ (██████) $█$ and ████████ at an █████ $\theta$ above the █-████, can be █████████ in █████████ ███████████ using the ████ ███████ ${\hat {\mathbf {x} }}$ and ${\hat {\mathbf {y} }}$ : ${\textbf {r}}=r\cos(\theta ){\hat {\mathbf {x} }}+r\sin(\theta ){\hat {\mathbf {y} }}.$

The ██████████ of ███████ ████████ ██████ ████████ █████ ██████:

The ██████ █████ ████ on a ██████.
The ██████ of the ██████ $█$ ████ not ██████ in ████.
The ██████ █████ with ████████ ███████ ████████ $\omega$ around the ██████. █████████, $\theta =\omega t$ █████ $█$ is ████.

The ████████ ${\textbf {v}}$ and ████████████ ${\textbf {a}}$ of the ██████ are the █████ and ██████ ███████████ of ████████ with ███████ to ████:

${\textbf {r}}=r\cos(\omega t){\hat {\mathbf {x} }}+r\sin(\omega t){\hat {\mathbf {y} }},$ ${\textbf {v}}={\dot {\textbf {r}}}=-r\omega \sin(\omega t){\hat {\mathbf {x} }}+r\omega \cos(\omega t){\hat {\mathbf {y} }},$ ${\textbf {a}}={\ddot {\textbf {r}}}=-\omega ^{2}(r\cos(\omega t){\hat {\mathbf {x} }}+r\sin(\omega t){\hat {\mathbf {y} }}).$

The ████ in ███████████ is the ████████ ██████████ of ${\textbf {r}}$ in █████████ ███████████. ████████████, ${\textbf {a}}=-\omega ^{2}{\textbf {r}}.$ The ████████ ████ █████ that the ████████████ is ███████ ███████ the ██████ of the ██████ (opposite the ██████), █████ it is ██████ "███████████" (█.█. "██████-███████"). █████ ███████ █████████ ██████ a ████████ ████ (███ to ███████), this ███████████ ████████████ █████████ the ████████ ██████ ████ ██████ by a ███████████ █████.

██████████ using ███████

The █████ at █████ █████ the ██████ █████████████ for ███████ ████████ ██████. The ████████ ██████ is ███████████ by the ███████ ████████ ██████ █, which is ██████ to the █████ of the █████ (using the █████-████ ████) and has █████████ █████ by:

|\mathbf {\Omega } |={\frac {\mathrm {d} \theta }{\mathrm {d} t}}=\omega \ ,

with █ the ███████ ████████ at ████ █. In this ██████████, ██/██ is ███████ ████████, ███████████ of ████. The ████████ ████████ ██ of the ████████ in ████ ██ along the ████████ ████ is

\mathrm {d} {\boldsymbol {\ell }}=\mathbf {\Omega } \times \mathbf {r} (t)\mathrm {d} t\ ,

which, by ██████████ of the ██████ █████ ███████, has █████████ ███ and is in the █████████ ███████ to the ████████ ████.

████████████,

{\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}=\lim _{{\Delta }t\to 0}{\frac {\mathbf {r} (t+{\Delta }t)-\mathbf {r} (t)}{{\Delta }t}}={\frac {\mathrm {d} {\boldsymbol {\ell }}}{\mathrm {d} t}}\ .

In █████ █████,

\mathbf {v} \ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}={\frac {\mathrm {d} \mathbf {\boldsymbol {\ell }} }{\mathrm {d} t}}=\mathbf {\Omega } \times \mathbf {r} (t)\ .

███████████████ with ███████ to ████, $\mathbf {a} \ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathrm {d} \mathbf {v} }{d\mathrm {t} }}=\mathbf {\Omega } \times {\frac {\mathrm {d} \mathbf {r} (t)}{\mathrm {d} t}}=\mathbf {\Omega } \times \left[\mathbf {\Omega } \times \mathbf {r} (t)\right]\ .$

████████'s ███████ ██████: $\mathbf {a} \times \left(\mathbf {b} \times \mathbf {c} \right)=\mathbf {b} \left(\mathbf {a} \cdot \mathbf {c} \right)-\mathbf {c} \left(\mathbf {a} \cdot \mathbf {b} \right)\ .$

████████ ████████'s ███████ with the ███████████ that █ • █(█) = █ at ███ █████, $\mathbf {a} =-{|\mathbf {\Omega |} }^{2}\mathbf {r} (t)\ .$

In █████, the ████████████ is ████████ ████████ opposite to the ██████ ████████████ █ at ███ █████, and has a █████████: $|\mathbf {a} |=|\mathbf {r} (t)|\left({\frac {\mathrm {d} \theta }{\mathrm {d} t}}\right)^{2}=r{\omega }^{2}$ █████ ████████ ████ |...| ██████ the ██████ █████████, which in the ████ of █(█) is ██████ the ██████ █ of the ████. This ██████ ██████ with the ████████ ███████, ██████ the ████████ is ████████ █████████.

████ the ████ of ████████ is ████ ████████ in the ████████ of ██████████ ████████ ██████, that ████████ ██████ with this ███.

A █████ of the ██████ ████████ is that it is ██████████ ███████████ of ███ ██████████ ██████.

███████: The ██████ ████

The █████ █████ in the █████ at █████ █████ a ████ in ████████ ██████ on a ██████ █████. The █████ is ██████ at an █████ █ from the ██████████, and the ███████ of the ████ is ██████████ to be ████████. The █████████ is to ████ ████ █████ the ████ ████ have ██ the ████ ████ not █████ off the ████. █████████ █████ ██ that, on a ████ █████ with ██ ███████ at ███, the ████ will ██████ █████ off the ████; █████ with a ████ █████ ███████, the ████ will █████ to the ██████ ██████ it ███████ the █████ ███████.

█████ from ███ ████████████ that █████ █████ in the █████████ of the ████, the █████ █████ of the █████ above █████████ the ██████ on the ████. There are ███ ██████; ███ is the █████ of ███████ ██████████ ████████ through the ██████ of ████ of the ████ ██, █████ █ is the ████ of the ████ and █ is the █████████████ ████████████; the ██████ is the ██████ ██████ █████ ███████ by the ████ at a █████ █████ to the ████ ███████ █a_█. The ███████████ █████ ████████ by the ██████ ██████ is ████ █████ above. This ███████████ █████ is not a █████ █████ ███████ to the ████, but ██████ ████ be ████████ by the ███ █████ on the ████ █████████ from ██████ ████████ of the ██████ █████ and the █████ of ███████. The █████████ or ███ █████ on the ████ █████ by ██████ ████████ of the ██████ █████ ███████ by the ████ and ████████ █████ ███ to ███████ ████ █████ the ███████████ █████ ████████ by the ████ to ██████ a ████████ ████. The ██████ ██████ is ██████████ ██ ████ as this ███ █████ ████████ the ███████████ █████ █████████ to the ██████.

The ██████████ ███ █████ on the ████ is the ██████████ █████████ of the █████ from the ████, which has █████████ |█_█| = █|a_█| ███ █. The ████████ █████████ of the █████ from the ████ ████ ██████████ the █████████████ █████: |█_█| = █|a_█| ███ █ = █|█|, which ███████ |a_█| = |█| / ███ █. ████████████ into the above ███████ for |█_█| ██████ a ██████████ █████ to be: $|\mathbf {F} _{\mathrm {h} }|=m|\mathbf {g} |{\frac {\sin \theta }{\cos \theta }}=m|\mathbf {g} |\tan \theta \,.$

On the █████ ████, at ████████ |█| on a ████████ ████ of ██████ █, ██████████ ████ that the █████ ██████ to ████ the ████ ████████████ into the ████ is the ████████ ██████ ███████████ █████ █_█ of █████████: $|\mathbf {F} _{\mathrm {c} }|=m|\mathbf {a} _{\mathrm {c} }|={\frac {m|\mathbf {v} |^{2}}{r}}\,.$

████████████, the ████ is in a ██████ ████ ████ the █████ of the ████ is ███ to ███████ the █████████: $m|\mathbf {g} |\tan \theta ={\frac {m|\mathbf {v} |^{2}}{r}}\,,$ or, $\tan \theta ={\frac {|\mathbf {v} |^{2}}{|\mathbf {g} |r}}\,.$

As the █████ of ████ █ ██████████ ██°, the ███████ ████████ ██████████ ████████, ████████ ██████ ██████ for |█|^█/█. In █████, this ████████ ██████ that for ███████ ██████ (██████ |█|) the ████ ████ be ██████ ████ ███████ (a ██████ █████ for █), and for ███████ █████ (███████ █) the ████ ████ ████ be ██████ ████ ███████, which ███████ with █████████. ████ the █████ █ ████ not ███████ the above █████████, the ██████████ █████████ of █████ ███████ by the ████ ████ not ███████ the ███████ ███████████ █████, and an ██████████ ██████████ █████ ██████████ to the ████ ███████ is ██████ upon to ███████ the ██████████. If ████████ ██████ do this (that is, the ███████████ of ████████ is ████████), the ████ ██████ to a █████████ ██████ █████ the ███████ can be ████████.

█████ █████ █████ to ███ ██████ as ████. ███ the ███ █████'s ██████.

██████████ ████████ ██████

As a ██████████████ of the ███████ ████████ ██████ ████, ███████ the ███████ ████ of ████████ is not ████████. The ████████████ ███ has a ██████████ █████████, as █████ the █████ at █████. This ████ is used to ███████████ a ██████████ ████████ █████ on a █████ ██████████ ██████.

███ █(█) be a ██████ that █████████ the ████████ of a █████ ████ as a ████████ of ████. Since ██ are ████████ ████████ ██████, ███ █(█) = █·█_█, █████ █ is a ████████ (the ██████ of the ██████) and █_█ is the ████ ██████ ████████ from the ██████ to the █████ ████. The █████████ of █_█ is █████████ by █, the █████ between the █-████ and the ████ ██████, ████████ ████████████████ from the █-████. The █████ ████ ██████ for █████ ███████████, █_█ is █████████████ to █_█ and ██████ in the █████████ of ██████████ █. █████ █████ ████ ███████ can be █████████ in █████ of █████████ ████ ███████ in the █ and █ ██████████, ███████ ${\hat {\mathbf {i} }}$ and ${\hat {\mathbf {j} }}$ ████████████: $\mathbf {u} _{r}=\cos \theta \ {\hat {\mathbf {i} }}+\sin \theta \ {\hat {\mathbf {j} }}$ and $\mathbf {u} _{\theta }=-\sin \theta \ {\hat {\mathbf {i} }}+\cos \theta \ {\hat {\mathbf {j} }}.$

███ can █████████████ to ████ ████████: ${\begin{aligned}\mathbf {v} &=r{\frac {d\mathbf {u} _{r}}{dt}}\\&=r{\frac {d}{dt}}\left(\cos \theta \ {\hat {\mathbf {i} }}+\sin \theta \ {\hat {\mathbf {j} }}\right)\\&=r{\frac {d\theta }{dt}}{\frac {d}{d\theta }}\left(\cos \theta \ {\hat {\mathbf {i} }}+\sin \theta \ {\hat {\mathbf {j} }}\right)\\&=r{\frac {d\theta }{dt}}\left(-\sin \theta \ {\hat {\mathbf {i} }}+\cos \theta \ {\hat {\mathbf {j} }}\right)\\&=r{\frac {d\theta }{dt}}\mathbf {u} _{\theta }\\&=\omega r\mathbf {u} _{\theta }\end{aligned}}$ █████ █ is the ███████ ████████ ██/██.

This ██████ for the ████████ ███████ ████████████ that the ████████ ██████ be ████████ ████████████ to the ██████, and that the █████████ of the ████████ ██████ be ██. ███████████████ █████, and ██████ that ${\frac {d\mathbf {u} _{\theta }}{dt}}=-{\frac {d\theta }{dt}}\mathbf {u} _{r}=-\omega \mathbf {u} _{r}\ ,$ ██ ████ that the ████████████, a is: $\mathbf {a} =r\left({\frac {d\omega }{dt}}\mathbf {u} _{\theta }-\omega ^{2}\mathbf {u} _{r}\right)\ .$

████, the ██████ and ██████████ ██████████ of the ████████████ are: $\mathbf {a} _{r}=-\omega ^{2}r\ \mathbf {u} _{r}=-{\frac {|\mathbf {v} |^{2}}{r}}\ \mathbf {u} _{r}$ and $\mathbf {a} _{\theta }=r\ {\frac {d\omega }{dt}}\ \mathbf {u} _{\theta }={\frac {d|\mathbf {v} |}{dt}}\ \mathbf {u} _{\theta }\ ,$ █████ |█| = █ █ is the █████████ of the ████████ (the █████).

█████ █████████ ███████ ██████████████ that, in the ████ of an ██████ that █████ along a ████████ ████ with a ████████ █████, the ████████████ of the ████ ███ be ██████████ into a █████████████ █████████ that ███████ the █████████ of ██████ (the ███████████ ████████████), and a ████████, or ██████████ █████████, that ███████ the █████.

███████ ██████ ██████

█████ ███████████

The above ███████ can be ███████ ███████ ████ ██████ in █████ ███████████, and at the ████ ████ ████████ to ███████ ██████ within a █████, as █████ next. █████ ███████████ in the █████ ██████ a ██████ ████ ██████ █_█ and an ███████ ████ ██████ █_█, as █████ above. A ████████ at ████████ █ is █████████ by:

$\mathbf {r} =\rho \mathbf {u} _{\rho }\ ,$

█████ the ████████ █ is used to ████████ the ████████ of the ████ from the ██████ ███████ of █ to █████████ that this ████████ is not █████, but ██████ with ████. The ████ ██████ █_█ ███████ with the ████████ and ██████ ██████ in the ████ █████████ as █(█). ████ ██████ █_█ ████ ███████ with the ████████ and █████ ██████████ to █_█. ████, █_█ and █_█ ████ a █████ █████████ ██████████ ██████ ████████ to the ████████, and ████ to the ████ █████████ by the ████████. By ██████ the ████ ███████ ██ █████ █████ ████████, as ████ in the ██████ at the ████ of the █████ above, it is ████ that █_█ and █_█ ████ a █████-██████ ████ with ████ on the ████ ██████ that █████ ████ and █████ on the █████████ of this ██████ with the ████ █████ █(█) as █(█).

████ the ████████ █████, ███ ████████ is

\mathbf {v} ={\frac {\mathrm {d} \rho }{\mathrm {d} t}}\mathbf {u} _{\rho }+\rho {\frac {\mathrm {d} \mathbf {u} _{\rho }}{\mathrm {d} t}}\,.

To ████████ the ████████, the ██████████ of the ████ ██████ █_█ is ██████. Because █_█ is a ████ ██████, ███ █████████ is █████, and it can ██████ ████ in █████████, that is, ███ ██████ ██_█ has a █████████ ████ █████████████ to █_█. ████ the ██████████ █(█) ███████ an ██████ ██, █_█, which ██████ in the ████ █████████ as █(█), ████ ███████ by ██. ███ █████ above. █████████, the ██████ in █_█ is

\mathrm {d} \mathbf {u} _{\rho }=\mathbf {u} _{\theta }\mathrm {d} \theta \,,

{\frac {\mathrm {d} \mathbf {u} _{\rho }}{\mathrm {d} t}}=\mathbf {u} _{\theta }{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\,.

In a ███████ ███████, the ████ of ██████ of █_█ is █████. As with █_█, █_█ is a ████ ██████ and can ████ ██████ without ████████ ████. To ██████ ██████████ to █_█ █████ the ██████████ █(█) ███████ an ██████ ██, █_█, which is ██████████ to █(█), ████ ███████ by ██. ███ █████ above. █████████, the ██████ ██_█ is ██████████ to █_█ and ████████████ to ██ (███ █████ above):

{\frac {\mathrm {d} \mathbf {u} _{\theta }}{\mathrm {d} t}}=-{\frac {\mathrm {d} \theta }{\mathrm {d} t}}\mathbf {u} _{\rho }\,.

The ████████ above █████ the ████ to be ████████: to ████████ █████████████, if ██_█ is ████████ with ██, ████ ██_█ ████ ████████.

████████████ the ██████████ of █_█ into the ██████████ for ████████:

\mathbf {v} ={\frac {\mathrm {d} \rho }{\mathrm {d} t}}\mathbf {u} _{\rho }+\rho \mathbf {u} _{\theta }{\frac {\mathrm {d} \theta }{\mathrm {d} t}}=v_{\rho }\mathbf {u} _{\rho }+v_{\theta }\mathbf {u} _{\theta }=\mathbf {v} _{\rho }+\mathbf {v} _{\theta }\,.

To ██████ the ████████████, ███████ ████ ███████████████ is ████:

\mathbf {a} ={\frac {\mathrm {d} ^{2}\rho }{\mathrm {d} t^{2}}}\mathbf {u} _{\rho }+{\frac {\mathrm {d} \rho }{\mathrm {d} t}}{\frac {\mathrm {d} \mathbf {u} _{\rho }}{\mathrm {d} t}}+{\frac {\mathrm {d} \rho }{\mathrm {d} t}}\mathbf {u} _{\theta }{\frac {\mathrm {d} \theta }{\mathrm {d} t}}+\rho {\frac {\mathrm {d} \mathbf {u} _{\theta }}{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}+\rho \mathbf {u} _{\theta }{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\,.

████████████ the ███████████ of █_█ and █_█, the ████████████ of the ████████ is:

{\begin{aligned}\mathbf {a} &={\frac {\mathrm {d} ^{2}\rho }{\mathrm {d} t^{2}}}\mathbf {u} _{\rho }+2{\frac {\mathrm {d} \rho }{\mathrm {d} t}}\mathbf {u} _{\theta }{\frac {\mathrm {d} \theta }{\mathrm {d} t}}-\rho \mathbf {u} _{\rho }\left({\frac {\mathrm {d} \theta }{\mathrm {d} t}}\right)^{2}+\rho \mathbf {u} _{\theta }{\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\ ,\\&=\mathbf {u} _{\rho }\left[{\frac {\mathrm {d} ^{2}\rho }{\mathrm {d} t^{2}}}-\rho \left({\frac {\mathrm {d} \theta }{\mathrm {d} t}}\right)^{2}\right]+\mathbf {u} _{\theta }\left[2{\frac {\mathrm {d} \rho }{\mathrm {d} t}}{\frac {\mathrm {d} \theta }{\mathrm {d} t}}+\rho {\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\right]\\&=\mathbf {u} _{\rho }\left[{\frac {\mathrm {d} v_{\rho }}{\mathrm {d} t}}-{\frac {v_{\theta }^{2}}{\rho }}\right]+\mathbf {u} _{\theta }\left[{\frac {2}{\rho }}v_{\rho }v_{\theta }+\rho {\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {v_{\theta }}{\rho }}\right]\,.\end{aligned}}

As a ██████████ ███████, if the ████████ █████ in a ██████ of ████████ ██████ █, ████ ██/██ = █, █ = █_█, and:

$\mathbf {a} =\mathbf {u} _{\rho }\left[-\rho \left({\frac {\mathrm {d} \theta }{\mathrm {d} t}}\right)^{2}\right]+\mathbf {u} _{\theta }\left[\rho {\frac {\mathrm {d} ^{2}\theta }{\mathrm {d} t^{2}}}\right]=\mathbf {u} _{\rho }\left[-{\frac {v^{2}}{r}}\right]+\mathbf {u} _{\theta }\left[{\frac {\mathrm {d} v}{\mathrm {d} t}}\right]\$

█████ $v=v_{\theta }.$

█████ ███████ █████ with █████ above for ██████████ ████████ ██████. ███ ████ the ███████ on ███-███████ ████████ ██████. If this ████████████ is ██████████ by the ████████ ████, the ███████ ████ is the ███████████ █████ and the ████████ of the ██████ ████ ███████ to ███████ ████████████ is █████████ ██████ the █████ █████.

For ████████████ █████ than ████████ ██████, for ███████, the ████ ███████ ██████████ ██████████ in the █████ above, the █████████████ ██████ of ████████ and ██████ of █████████ of the ██████████ are ███████ ████ ██████████ to the ██████████ ██████ ███████ by █_█ and █_█ and to the ██████ |█(█)| = █. ████████████, in the ███████ ████, it is not ███████████████ to ███████████ the ███████████ and █████ █████ from the above ███████ ████████████ ████████. To ████ ████████ with this █████, █████ ███████████ are ██████████, as █████████ next.

█████ ███████████

█████ ███████████ ████ a ███ of ███████████ that ██████ with the ████████, and have ███████████ ██████████ by the ████ of the ████████. ████ ███████ are ██████ as █████ in the █████ at █████, ████ ██████████ and ██████ to the ████. This ██████████ ██████ █████████ is ████████ to as █████████ or ████ ███████████ or ██-███████████, for ██████-██████████, █████████ to █████ ████ ███████. █████ ███████████ are a ████ ███████ ███████ of a ████ ███████ ███████ of █████ ███████████ from the ██████ of ████████████ █████.

████████ along the ████ of the ████████ is the ███ ██████ █, ██████████ to be a █████ ████████ of ████.

s=s(t)\ .

A ██████ of █████████ is ███████ at ████ ████████ █ ███████ a ████████ █ (the ██████ of █████████) from the █████ on a ████ along the ██████ █_█ (█). The ████████ ████████ █(█) at ███ ██████ █ is ███████ in █████ of the ████ of ████████ of the ███████ to the █████, which in ████ is ██████████ by the ████ ██████. If the ███████████ of the ███████ ████████ to some ████████ ████████ is █(█), ████ █(█) is ███████ by the ██████████ ██/██:

{\frac {1}{\rho (s)}}=\kappa (s)={\frac {\mathrm {d} \theta }{\mathrm {d} s}}\ .

The ██████ of █████████ ███████ is █████ as ████████ (that is, as an ████████ █████), █████ the █████████ █ is a ██████ ████████.

A █████████ ████████ to ███████ the ██████ of █████████ and the ██████ of █████████ uses a ████████ ███████ ███████ to the ██████████ ██████. ███ █████ above.

Using █████ ███████████, the ██████ along the ████ is ██████ as a ██████████ of ████████ █████ of ████-████████ ██████, and at ████ ████████ █ ███████████ ███-███████ ████████ ██████ at that ████████ with ██████ █. The █████ █████ of the ███████ ████ of ████████ ████ is █████ by:

\omega (s)={\frac {\mathrm {d} \theta }{\mathrm {d} t}}={\frac {\mathrm {d} \theta }{\mathrm {d} s}}{\frac {\mathrm {d} s}{\mathrm {d} t}}={\frac {1}{\rho (s)}}\ {\frac {\mathrm {d} s}{\mathrm {d} t}}={\frac {v(s)}{\rho (s)}}\ ,

with the █████ █████ █ █████ by:

v(s)={\frac {\mathrm {d} s}{\mathrm {d} t}}\ .

As for the █████ ████████ above, because ████ ███████ ██████ ██████ █████████, █████ ████ of ██████ is ██████ █████████████ to █████ █████████ (███ the ████-████ ██████ in the █████ above):

{\frac {d\mathbf {u} _{\mathrm {n} }(s)}{ds}}=\mathbf {u} _{\mathrm {t} }(s){\frac {d\theta }{ds}}=\mathbf {u} _{\mathrm {t} }(s){\frac {1}{\rho }}\ ;

;}

{\frac {d\mathbf {u} _{\mathrm {t} }(s)}{\mathrm {d} s}}=-\mathbf {u} _{\mathrm {n} }(s){\frac {\mathrm {d} \theta }{\mathrm {d} s}}=-\mathbf {u} _{\mathrm {n} }(s){\frac {1}{\rho }}\ .

████████████, the ████████ and ████████████ are:

\mathbf {v} (t)=v\mathbf {u} _{\mathrm {t} }(s)\ ;

;}

and using the █████-████ of ███████████████:

\mathbf {a} (t)={\frac {\mathrm {d} v}{\mathrm {d} t}}\mathbf {u} _{\mathrm {t} }(s)-{\frac {v^{2}}{\rho }}\mathbf {u} _{\mathrm {n} }(s)\ ;

;} with the ██████████ ████████████

{\frac {\mathrm {\mathrm {d} } v}{\mathrm {\mathrm {d} } t}}={\frac {\mathrm {d} v}{\mathrm {d} s}}\ {\frac {\mathrm {d} s}{\mathrm {d} t}}={\frac {\mathrm {d} v}{\mathrm {d} s}}\ v\ .

In this █████ ██████████ ██████, the ████████████ █████████ the ██████████ for ██████████ ████████ ██████ with the █████ ██████ █(█), and the ███████████ ████████████ is ██████████ as the ██████ ████.

█████████ this ████████ to █████ ███████████ █████ ██████ █████ to the ██████–██████ ████████.

███████████ ████████

███████ at the █████ above, ███ █████ ██████ ███████ ████████ ███████ has been █████ of the ██████████ in █████████ between █(█) and █(█ + ██) in █████████ the ███ ██████ as ██ = █(█)██. ███████████ on this █████ can be █████ using a ████ ██████ ████████ ████████ below. This ████████ ████ █████ ██████████ with the ███████ on █████████.

To █████████ the ████ ███████ of the █████ ██████████ ██████, ███ ████████ is to █████ in █████████ ███████████ and ████████ the █████ ███████████ in █████ of █████ █████████ ███████████. In █████ of ███ ██████ █, ███ the ████ be █████████ as: $\mathbf {r} (s)=\left[x(s),\ y(s)\right].$

████ an ███████████ ████████████ along the ████ ██ is █████████ by: $\mathrm {d} \mathbf {r} (s)=\left[\mathrm {d} x(s),\ \mathrm {d} y(s)\right]=\left[x'(s),\ y'(s)\right]\mathrm {d} s\ ,$

█████ ██████ are ██████████ to ██████ ███████████ with ███████ to █. The █████████ of this ████████████ is ██, ███████ that:

\left[x'(s)^{2}+y'(s)^{2}\right]=1\ .

(██. █)

This ████████████ is ███████████ a ███████ to the █████ at █, ███████ that the ████ ██████ ███████ to the █████ is: $\mathbf {u} _{\mathrm {t} }(s)=\left[x'(s),\ y'(s)\right],$ █████ the ███████ ████ ██████ ██████ to the █████ is $\mathbf {u} _{\mathrm {n} }(s)=\left[y'(s),\ -x'(s)\right],$

█████████████ can be ████████ by ███████ that the ██████ ███ ███████ is ████. The ████ █████████ of █████ ███████ is a ███████████ of ██. █. Using the ███████ ██████, the █████ █ of the ███████ to the █████ is █████ by: $\sin \theta ={\frac {y'(s)}{\sqrt {x'(s)^{2}+y'(s)^{2}}}}=y'(s)\ ;$ ;} and $\cos \theta ={\frac {x'(s)}{\sqrt {x'(s)^{2}+y'(s)^{2}}}}=x'(s)\ .$

The ██████ of █████████ is ██████████ ██████████ ████████ (without ████ for █████████ ██████████████) as: ${\frac {1}{\rho }}={\frac {\mathrm {d} \theta }{\mathrm {d} s}}\ .$

The ██████████ of █ can be █████ from that for ████: ${\frac {\mathrm {d} \sin \theta }{\mathrm {d} s}}=\cos \theta {\frac {\mathrm {d} \theta }{\mathrm {d} s}}={\frac {1}{\rho }}\cos \theta \ ={\frac {1}{\rho }}x'(s)\ .$

███: ${\frac {\mathrm {d} \sin \theta }{\mathrm {d} s}}={\frac {\mathrm {d} }{\mathrm {d} s}}{\frac {y'(s)}{\sqrt {x'(s)^{2}+y'(s)^{2}}}}={\frac {y''(s)x'(s)^{2}-y'(s)x'(s)x''(s)}{\left(x'(s)^{2}+y'(s)^{2}\right)^{3/2}}}\ ,$ in which the ███████████ is █████. With this ███████ for the ██████████ of the ████, the ██████ of █████████ ███████: ${\frac {\mathrm {d} \theta }{\mathrm {d} s}}={\frac {1}{\rho }}=y''(s)x'(s)-y'(s)x''(s)={\frac {y''(s)}{x'(s)}}=-{\frac {x''(s)}{y'(s)}}\ ,$ █████ the ███████████ of the █████ █████ from ███████████████ of ██. █: $x'(s)x''(s)+y'(s)y''(s)=0\ .$ With █████ ███████, the ████████████ can be █████: ${\begin{aligned}\mathbf {a} (s)&={\frac {\mathrm {d} }{\mathrm {d} t}}\mathbf {v} (s)={\frac {\mathrm {d} }{\mathrm {d} t}}\left[{\frac {\mathrm {d} s}{\mathrm {d} t}}\left(x'(s),\ y'(s)\right)\right]\\&=\left({\frac {\mathrm {d} ^{2}s}{\mathrm {d} t^{2}}}\right)\mathbf {u} _{\mathrm {t} }(s)+\left({\frac {\mathrm {d} s}{\mathrm {d} t}}\right)^{2}\left(x''(s),\ y''(s)\right)\\&=\left({\frac {\mathrm {d} ^{2}s}{\mathrm {d} t^{2}}}\right)\mathbf {u} _{\mathrm {t} }(s)-\left({\frac {\mathrm {d} s}{\mathrm {d} t}}\right)^{2}{\frac {1}{\rho }}\mathbf {u} _{\mathrm {n} }(s)\end{aligned}}$ as can be ████████ by ██████ the ███ ███████ with the ████ ███████ █_█(█) and █_█(█). This ██████ for ████████████ is the ████ as that for ████████ ██████ █████ on the ██████ █. Using this ██████████ ██████ in the ████████ █████, it is ████ to ████████ the █████ ██████ to the ██████████ as the ███████████ █████ and that ████████ to the ██████████ as the ██████████ █████. From a ███████████ ██████████, the ████ can be ████████████ by an ███ of a ██████ for a ███████ ████, and for the ███████ ████ a ██████████ ██████ of █████████ ███████, the ███████████ and █████ ██████ can be ████████ on the █████ of ████████ ██████ with that ██████.

This ██████ for ████████████ ██████ with that █████ ███████. ███████, in this ████████, the ████████ of the ██████ in ██████ of █████████ with █ is ███████ ██████████ ████████, ██████████ with a █████████ ██████████████, but not ███████ upon it, ███████ ████████ ███ █████████ the █████ above █████ ███████ about ██████████ the █████████ in █.

███████: ████████ ██████

To ██████████ the above ████████, ███ █, █ be █████ as:

x=\alpha \cos {\frac {s}{\alpha }}\ ;\ y=\alpha \sin {\frac {s}{\alpha }}\ .

;\ █=\█████ \███ {\████ {█}{\█████ }}\ .}

████:

x^{2}+y^{2}=\alpha ^{2}\ ,

which can be ██████████ as a ████████ ████ around the ██████ with ██████ █. The ████████ █ = █ ███████████ to [█, █], or █ █'█████. To use the above █████████, the ███████████ are ██████:

y^{\prime }(s)=\cos {\frac {s}{\alpha }}\ ;\ x^{\prime }(s)=-\sin {\frac {s}{\alpha }}\ ,

;\ █^{\█████ }(█)=-\███ {\████ {█}{\█████ }}\ ,}

y^{\prime \prime }(s)=-{\frac {1}{\alpha }}\sin {\frac {s}{\alpha }}\ ;\ x^{\prime \prime }(s)=-{\frac {1}{\alpha }}\cos {\frac {s}{\alpha }}\ .

;\ █^{\█████ \█████ }(█)=-{\████ {█}{\█████ }}\███ {\████ {█}{\█████ }}\ .}

With █████ ███████, ███ can ██████ that:

x^{\prime }(s)^{2}+y^{\prime }(s)^{2}=1\ ;\ {\frac {1}{\rho }}=y^{\prime \prime }(s)x^{\prime }(s)-y^{\prime }(s)x^{\prime \prime }(s)={\frac {1}{\alpha }}\ .

;\ {\████ {█}{\███ }}=█^{\█████ \█████ }(█)█^{\█████ }(█)-█^{\█████ }(█)█^{\█████ \█████ }(█)={\████ {█}{\█████ }}\ .}

The ████ ███████ can ████ be █████:

\mathbf {u} _{\mathrm {t} }(s)=\left[-\sin {\frac {s}{\alpha }}\ ,\ \cos {\frac {s}{\alpha }}\right]\ ;\ \mathbf {u} _{\mathrm {n} }(s)=\left[\cos {\frac {s}{\alpha }}\ ,\ \sin {\frac {s}{\alpha }}\right]\ ,

;\ \██████ {█} █{\██████ {█} }(█)=\████[\███ {\████ {█}{\█████ }}\ ,\ \███ {\████ {█}{\█████ }}\█████]\ ,}

which █████ to ████ that █ = █ is ███████ at ████████ [█, █] and █ = ██/█ at [█, █], which ██████ with the ████████ ███████████ for █ and █. In █████ █████, █ is ████████ ████████████████ around the ██████ from █ █'█████. ████, the ███████████ of █████ ███████ can be █████:

{\frac {\mathrm {d} }{\mathrm {d} s}}\mathbf {u} _{\mathrm {t} }(s)=-{\frac {1}{\alpha }}\left[\cos {\frac {s}{\alpha }}\ ,\ \sin {\frac {s}{\alpha }}\right]=-{\frac {1}{\alpha }}\mathbf {u} _{\mathrm {n} }(s)\ ;

;}

\ {\frac {\mathrm {d} }{\mathrm {d} s}}\mathbf {u} _{\mathrm {n} }(s)={\frac {1}{\alpha }}\left[-\sin {\frac {s}{\alpha }}\ ,\ \cos {\frac {s}{\alpha }}\right]={\frac {1}{\alpha }}\mathbf {u} _{\mathrm {t} }(s)\ .

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s(t)=\int _{0}^{t}\ dt^{\prime }\ v(t^{\prime })\ ,

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\mathbf {v} =v(t)\mathbf {u} _{\mathrm {t} }(s)\ ,

\mathbf {a} ={\frac {\mathrm {d} v}{\mathrm {d} t}}\mathbf {u} _{\mathrm {t} }(s)+v{\frac {\mathrm {d} }{\mathrm {d} t}}\mathbf {u} _{\mathrm {t} }(s)={\frac {\mathrm {d} v}{\mathrm {d} t}}\mathbf {u} _{\mathrm {t} }(s)-v{\frac {1}{\alpha }}\mathbf {u} _{\mathrm {n} }(s){\frac {\mathrm {d} s}{\mathrm {d} t}}

\mathbf {a} ={\frac {\mathrm {d} v}{\mathrm {d} t}}\mathbf {u} _{\mathrm {t} }(s)-{\frac {v^{2}}{\alpha }}\mathbf {u} _{\mathrm {n} }(s)\ ,

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