Question

A toy cart is tied to the end of an unstretched string of length '$l$'. When revolved, the toy card moves in horizontal circle with radius '$2l$' and time period $T$. If it is speeded until it moves in horizontal circle of radius '$3l$' with period $T_1$, relation between $T$ and $T_1$ is (Hooke's law is obeyed)

• A. $T_1=\frac{2}{\sqrt{3}}T$
• B. $T_1=\sqrt{\frac{3}{2}}T$
• C. $T_1=\sqrt{\frac{2}{3}}T$
• D. $T_1=\frac{\sqrt{3}}{2}T$

Answer 1

The correct option is D $T_1=\frac{\sqrt{3}}{2}T$

$T=\frac{2\pi r}{\omega }$ (Time period)

$\frac{F}{A}=Y.\frac{\Delta l}{l}$

Initially,$F=m{\omega }^{2}r$

$\Delta l=l$

$F=YA$

$YA=m{\omega }^{2}r=m{\omega }^2.2l$

$\omega =\sqrt{\frac{YA}{2ml}}$

$\Delta l=2l$

$F=2YA$

$2YA=m{\omega }_1^2.3l$

${\omega }_1=\sqrt{\frac{2YA}{3ml}}$

$T=\frac{2\pi }{\omega }$

$T_1=\frac{2\pi }{{\omega }_1}$

$\frac{T}{T_1}=\frac{\omega }{{\omega }_1}$

$=\sqrt{\frac{2YA\times 2ml}{3ml\times YA}}=\frac{2}{\sqrt{3}}$

$T_1=\frac{\sqrt{3}}{2}T$