Question

A cirlce with a radius of 8 cm rotates 30 degrees in one second. Determine the angle of rotation in radians.

Angle:___ w:___ v:___

Answer 1

same as before, since 180° is π, how much is 30° in radians?


`\bf \begin{array}{ccll} de grees&radians\\ \text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\ 180&\pi \\\\ 30&x \end{array}\implies \cfrac{180}{30}=\cfrac{\pi }{x}\implies x=\cfrac{30\pi }{180}\implies x=\cfrac{\pi }{6}\\\\ -------------------------------\\\\ \stackrel{angular~velocity}{w}=\cfrac{\stackrel{central~angle}{(\pi )/(6)}}{\stackrel{time}{1~s}}\implies w=\cfrac{\pi }{6}~(radians)/(seconds)`


`\bf \textit{arc's length}\\\\ s=r\theta ~~ \begin{cases} r=radius\\ \theta =angle~in\\ \qquad radians\\ ------\\ r=8\\ \theta =(\pi )/(6) \end{cases}\implies s=8\cdot \cfrac{\pi }{6}\implies s=\cfrac{4\pi }{3} \\\\\\ \stackrel{linear~velocity}{v}=\cfrac{\stackrel{arc's~length}{(4\pi )/(3)}}{\stackrel{time}{1~s}}\implies v=\cfrac{4\pi }{3}~(cm)/(seconds)`