Question

A wheel is subjected to uniform angular acceleration about its axis. Initially its angular velocity is zero. In the first $2$ seconds, it rotates through an angle ${\theta }_1$. In the next $2$ seconds, it rotates through an additional angle ${\theta }_2$. Find the ratio of ${\theta }_2/{\theta }_1$.

Answer 1

$Usingrelation\theta =w_{0}t+\frac{1}{2}a{t}^2............\left(i\right)hereweknowthat{w}_0=0,t=2sec\therefore {\theta }_1=\frac{1}{2}\left(\alpha \right)(2{)}^2=(2\alpha )Nowusingsameequationfort=4sec,w_0=0{\theta }_1+{\theta }_2=\frac{1}{2}\alpha (4{)}^2=8\alpha ...........\left(ii\right){\theta }_2=8\alpha -2\alpha =6\alpha From\left(i\right)and\left(ii\right),{\theta }_1=2\alpha ,{\theta }_2=6\alpha \therefore \frac{{\theta }_2}{{\theta }_1}=\frac{6\alpha }{2\alpha }=3$