Question

_{A PARTICLE IS MOVING ALONG A CIRCULAR PATH OF RADIUS R IN SUCH A WAY THAT AT ANY INSTANT MAGNITUDE OF RADIAL ACCELERATION AND TANGENTIAL ACCELERATION ARE EQUAL. IF t=0 VELOCITY OF PARTICLE IS Vo.THEN FIND THE TIME PERIOD OF FIRST REVOLUTION OF THE PARTICLE?}

Answer 1

Dear Student,

Radial and tangential acceleration given by -
$$\begin{gathered} a_t=\frac{dv}{dt},a_r=\frac{v^2}{r} \\ Accordingtoquestion \\ {a}_t=a_r \\ \frac{dv}{dt}=\frac{v^2}{r}\Rightarrow \frac{dv}{v^2}=\frac{dt}{r} \\ Integratingthis \\ \int \frac{dv}{v^2}=\int \frac{dt}{r} \\ -\frac{1}{v}=\frac{t}{R}+c \\ giveninquestionatt=o,v=v_0 \\ c=-\frac{1}{v_0} \\ \Rightarrow -\frac{1}{v}=\frac{t}{R}-\frac{1}{v_0} \\ t=R(\frac{1}{v_0}-\frac{1}{v}) \end{gathered}$$