Consider an object making uniform motion around a circle of radius $5 \mathrm{~m}$ with tangential velocity $2 \mathrm{~ms}^{-1}$. The time it takes to complete 2 revolution and the magnitude of acceleration respectively are

• A

  $0.2 \pi \mathrm{s}$ and $0.8 \mathrm{~ms}^{-2}$
• B

  $0.5 \pi \mathrm{s}$ and $1 \mathrm{~ms}^{-2}$
• Correct answer
  C

  $10 \pi \mathrm{s}$ and $0.8 \mathrm{~ms}^{-2}$
• D

  $5 \pi \mathrm{s}$ and $5 \mathrm{~ms}^{-2}$

Solution

Given that, circle of radius $=5 \mathrm{~m}$ tangential velocity $=2 \mathrm{~m} / \mathrm{s}$
$$ \text { Here, } 2 \text { revolutions complete by object. } $$ $\because$ We know that, $$ \begin{aligned} & v & =\omega R \Rightarrow \frac{v}{R}=\omega \\ \Rightarrow & \omega & =\frac{2}{5} \mathrm{rad} / \mathrm{s} \\ \text { Also, } & \omega & =\frac{2 \pi}{T} \Rightarrow T=\frac{2 \pi}{\omega}=\frac{2 \pi}{2 / 5} \end{aligned} $$ $T=5 \pi$ s for one revolution. $\Rightarrow$ The body completes 2 revolutions in time, $t=2 \times 5 \times \pi=10 \pi \mathrm{s}$ Also acceleration, $\alpha=v^2 / R=\frac{2^2}{5}=0.8 \mathrm{~ms}^{-2}$

Hence, the correct answer is option: 'C'.