Question

A particle starts from rest and moves in a circular motion with constant angular acceleration of $2rad/s^2$. Find
a. Angular velocity
b. Angular displacement of the particle after $4s$
c. The number of revolutions completed by the particle during these $4s$.
d. If the radius of the circle is $10cm$, find the magnitude and direction of net acceleration of the particle at the end of $4s$.

Answer 1

$\alpha =2rad{s}^{-2},{\omega }_1=0,t=4s$

a.)$\omega ={\omega }_1+\alpha t\Rightarrow {\omega }_2=0+2\times 4=8rad{s}^{-1}$

b.)$\theta ={\omega }_{1}t+\frac{1}{2}\alpha t^2\Rightarrow \theta =0\times 4+\frac{1}{2}\times 2\times 4^2=16rad$

c.) $1$ revolution $=2\pi rad\Rightarrow 1rad=\frac{1}{2}rev\Rightarrow 16rad=\frac{16}{2\pi }rev=\frac{8}{\pi }rev=\frac{8\times 7}{22}=\frac{28}{11}rev$

d.) After $4s:v={\omega }_{2}r=8\times 10/100=0.8m{s}^{-1}$

$a_c=\frac{v^2}{r}=\frac{(0.8{)}^2}{10/100}=6.4m{s}^{-}2$

Now magnitude of net acceleration,

$a=\sqrt{a_c^2+a_t^2}=\sqrt{{6.4}^2+{0.2}^2}=\sqrt{41}m{s}^{-2}$

Direstion on acceleration

$\tan \theta =\frac{a_t}{a_c}=\frac{0.2}{6.4}=\frac{1}{32}\Rightarrow \theta ={\tan }^{-1}\left(\frac{1}{32}\right)$