Question

A toy cart is tied to the end of an unstretched string of length 'I' when revolved, the toy cart moves in horizontalline with radius ${}^{\prime }2I^{\prime }$ and time period $T$ . If it is speeded unit moves in horizontal circle of radius ${}^{\prime }3I^{\prime }$ with period of relation between $T$ and $T_1$ is (Hooke's law is obeyed)

• A. $T_1=\frac{2}{\sqrt{3}}$
• 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

Answer: (B)

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

$\text{Case(1):}\Delta x=2I-I=I$

$\begin{array}{cc}R\Delta x& =\frac{m{v}^2}{r}\\ \Rightarrow v^2& =\sqrt{\frac{kr\left(\Delta x\right)}{m}}\\ & T=\frac{2\pi r}{v}\\ \Rightarrow T& =\frac{2\pi r\sqrt{m}}{\sqrt{kr\left(\Delta x\right)}}⟶\left(1\right)\end{array}$

$T^{}=4\pi I\sqrt{\frac{m}{K\left(2I\right)\left(I\right)}}=4\pi \sqrt{\frac{m}{2K}}=\pi \sqrt{\frac{m}{k}}\frac{4}{\sqrt{2}}$

$\begin{array}{c}\text{Case}\left(2\right)\text{using}\left(1\right)\\ \Delta x=2I\\ r=3I\\ T=2\pi \left(3I\right)\sqrt{\frac{m}{K\left(3I\right)\left(2I\right)}}\end{array}$

$\begin{array}{c}\Rightarrow T_1=6\pi \sqrt{\frac{m}{6k}}=\pi \sqrt{\frac{m}{k}}\sqrt{6}\\ \frac{T}{T_1}=\frac{4}{\sqrt{2}\sqrt{6}}=\frac{4}{\sqrt{12}}=\frac{4}{2\sqrt{3}}=\frac{2}{\sqrt{3}}\\ \Rightarrow T_1=\frac{\sqrt{3}}{2}T\text{,Ans: (B)}\end{array}$