Question

A toy cart is tied to the end of an unstretched string of length $l$. When revolved, the toy cart 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 $T_1=\frac{\sqrt{3}}{2}T$

$F=\frac{AY\Delta l}{l}$ .............(i)

$F=\frac{{mv}^2}{r}$.............(ii)

From (i) and (ii),

$v=\sqrt{AYr\Delta l/lm}$...........(iii)

$T=\frac{2\pi r}{v}$............(iv)

From (iii) and (iv),

$T=\frac{2\pi \sqrt{rlm}}{\sqrt{AY\Delta l}}$

$T\propto \sqrt{\frac{r}{\Delta l}}$

For $r=2l,\Delta l=l,T=T$

For $r=3l,\Delta l=2l,T=T_1$

$⟹\frac{T_1}{T}=\frac{\sqrt{3l/2l}}{\sqrt{l/2l}}$

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