Question

Find the rotational velocity
If Centripetal Acceleration = 2019 m/s²
and Radius = 06 m

Answer 1

Answer:

$v=3\sqrt{1346}~m/s\quad \begin{pmatrix}110.06362\dots~m/s\\396.22902\dots~km/h\end{pmatrix}$

Step-by-step explanation:

$\mathrm{Centripetal\:Acceleration}$

• $\mathrm{For\:Rotation\:Velocity=v,\:Radius=r}$
• $\mathrm{The\:Centripetal\:Acceleration\:Formula\:is:}\:a_c=\dfrac{v^2}{t}$

$a_c=2019$

$r=6$

$2019=\dfrac{v^2}{6}$

$2019=\dfrac{v^2}{6}$

$\mathrm{Switch\:sides}$

$\dfrac{v^2}{6}=2019$

$\mathrm{Multiply\:both\:sides\:by\:}6$

$\dfrac{6v^2}{6}=2019\cdot \:6$

$\mathrm{Simplify}$

$v^2=12114$

$\mathrm{For\:}x^2=f\left(a\right)\mathrm{\:the\:solutions\:are\:}x=\sqrt{f\left(a\right)},\:\:-\sqrt{f\left(a\right)}$

$v=\sqrt{12114},\:v=-\sqrt{12114}$

$\sqrt{12114}$

$\mathrm{Prime\:factorization\:of\:}12114$

$12114$$12114\:\mathrm{divides\:by}\:2\quad \:12114=6057\cdot \:2$

$=2\cdot \:6057$

$6057\:\mathrm{divides\:by}\:3\quad \:6057=2019\cdot \:3$

$=2\cdot \:3\cdot \:2019$

$2019\:\mathrm{divides\:by}\:3\quad \:2019=673\cdot \:3$

$=2\cdot \:3\cdot \:3\cdot \:673$

$2,\:3,\:673\mathrm{\:are\:all\:prime\:numbers,\:therefore\:no\:further\:factorization\:is\:possible}$

$=2\cdot \:3\cdot \:3\cdot \:673$

$=2\cdot \:3^2\cdot \:673$

$=\sqrt{3^2\cdot \:2\cdot \:673}$

$\mathrm{Apply\:radical\:rule}:\quad \sqrt[n]{ab}=\sqrt[n]{a}\sqrt[n]{b}$

$=\sqrt{3^2}\sqrt{2\cdot \:673}$

$\mathrm{Apply\:radical\:rule}:\quad \sqrt[n]{a^n}=a$

$\sqrt{3^2}=3$

$=3\sqrt{2\cdot \:673}$

$\mathrm{Refine}$

$=3\sqrt{1346}$

$\mathrm{Like\:wise\:}-\sqrt{12114}=-3\sqrt{1346}$

$v=3\sqrt{1346},\:v=-3\sqrt{1346}$

$\mathrm{Selecting\:Positive\:value}$

$v=3\sqrt{1346}$

$\mathrm{Units\:used\:for\:calculation:}\:Meters\:per\:Second$

$v=3\sqrt{1346}~m/s$