Question

Two bodies $A$ & $B$ rotate about an axis such that angle ${\theta }_A$ (in radians) covered by first body is proportional to square of time, & ${\theta }_B$ (in radians) covered by second body varies linearly. At $t=0,{\theta }_A={\theta }_B=0$. If $A$ completes its first revolution in $\sqrt{\pi }\text{s}$ & $B$ needs $4\pi \text{s}$ to complete half revolution, then angular velocities ${\omega }_A$ & ${\omega }_B$ at $t=5\text{s}$ are in the ratio

• A. $4:1$
• B. $20:1$
• C. $80:1$
• D. $20:4$

Answer 1

The correct option is C $80:1$

Let the two equations be
${\theta }_A=C_A{t}^2$
${\theta }_B=C_{B}t$

From given data
$2\pi =C_A\pi$
$C_A=2$ and
$\pi =C_{B}4\pi$
$C_B=\frac{1}{4}$

Therefore equations become
${\theta }_A=2t^2$
${\theta }_B=t/4$

${\omega }_A=\frac{d{\theta }_A}{dt}=\frac{d\left(2t^2\right)}{dt}=4t$
${\omega }_B=\frac{d{\theta }_B}{dt}=\frac{d\left(t/4\right)}{dt}=\frac{1}{4}$
${\omega }_{A{|}_5}=4\times 5=20$
$\Rightarrow {\omega }_A:{\omega }_B=20:\frac{1}{4}=80:1$