Aplūkosim riteņbraucēja kustību velotreka pagriezienā.

Centrtieces paātrinājums vērsts pa riņķa līnijas rādiusu uz tās centru. Kāds spēks izraisa centrtieces paātrinājumu? Tas ir miera berzes spēks, kurš rodas starp riepām un ceļa virsmu un pretojas sānslīdei, un reakcijas spēks, kurš vērsts perpendikulāri virsmai.

Pēc otrā Ņūtona likuma ${F}_{b}\cdot \mathit{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}{F}_{r}\cdot \mathit{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}m\cdot {a}_{c},\phantom{\rule{0.147em}{0ex}}\mathit{kur}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}{a}_{c}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{{v}^{2}}{R}\phantom{\rule{0.294em}{0ex}}\mathit{un}\phantom{\rule{0.294em}{0ex}}{F}_{b}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\mathrm{\mu }\cdot {F}_{r}$.

No tā izriet $\begin{array}{l}\mathrm{\mu }\cdot {F}_{r}\cdot \mathit{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}{F}_{r}\cdot \mathit{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}m\cdot \frac{{v}^{2}}{R}\\ {F}_{r}\cdot \left(\mathrm{\mu }\cdot \mathit{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\right)\phantom{\rule{0.294em}{0ex}}=\phantom{\rule{0.147em}{0ex}}m\cdot \frac{{v}^{2}}{R}\end{array}$ — projekcija uz $$x$$ ass.

Savukārt projekcija uz y ass atbilstoši 2.Ņūtona likumam:

$\begin{array}{l}{F}_{r}\cdot \mathit{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}{F}_{b}\cdot \mathit{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}{F}_{\mathit{sm}}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}0,\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\mathit{kur}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}{F}_{\mathit{sm}}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\mathit{mg}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\mathit{un}\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}{F}_{b}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\mathrm{\mu }\cdot {F}_{r}\\ {F}_{r}\cdot \mathit{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}\mathrm{\mu }\cdot {F}_{r}\cdot \mathit{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}\mathit{mg}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}0\\ {F}_{r}\cdot \left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}\mathrm{\mu }\cdot \mathit{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\right)\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}\mathit{mg}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}0\end{array}$

Apvienojot formulas, var iegūt $\begin{array}{l}\frac{\mathit{mg}}{\left(\mathit{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}\mathrm{\mu }\cdot \mathit{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\right)}\cdot \left(\mathrm{\mu }\cdot \mathit{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\right)\phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}m\cdot \frac{{v}^{2}}{R}\\ \frac{\mathrm{\mu }\cdot \mathit{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }}{\mathit{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}\mathrm{\mu }\cdot \mathit{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }}\phantom{\rule{0.147em}{0ex}}=\phantom{\rule{0.147em}{0ex}}\frac{{v}^{2}}{\mathit{gR}}\end{array}$

Lai nebūtu sānslīdes, pieļaujamais ātrums pagriezienā ir

$v\phantom{\rule{0.147em}{0ex}}\le \phantom{\rule{0.147em}{0ex}}\phantom{\rule{0.147em}{0ex}}\sqrt{\mathit{gR}\cdot \frac{\mathrm{\mu }\cdot \mathit{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}+\phantom{\rule{0.147em}{0ex}}\mathit{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }}{\mathit{cos}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }\phantom{\rule{0.147em}{0ex}}-\phantom{\rule{0.147em}{0ex}}\mathrm{\mu }\cdot \mathit{sin}\phantom{\rule{0.147em}{0ex}}\mathrm{\alpha }}}=\phantom{\rule{0.147em}{0ex}}\sqrt{\mathit{gR}\cdot \frac{\mathrm{\mu }\phantom{\rule{0.147em}{0ex}}+\mathit{tg}\mathrm{\alpha }}{1\phantom{\rule{0.147em}{0ex}}-\mathrm{\mu }\cdot \mathit{tg}\mathrm{\alpha }}}$.

Šis ātrums, nemainoties ceļa segumam un ceļa slīpuma leņķim pret horizontu, ir atkarīgs tikai no pagrieziena rādiusa $$R$$. Jo lielāks ir pagrieziena rādiuss, jo lielāks ir pieļaujamais kustības ātrums.