Question

A toy cart is tied the end of an unstrtched string of length a. when revolved, the toy cart moves in a horizontal circle of radius 2a with a time period T. now the toy cart is speeded up until it moves in a horizontal circle of radius 3a with a period T. if Hooke's law holds then

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

Answer 1

Answer: (C)

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

$\begin{array}{c}F=\frac{m{v}^2}{R}-\left(1\right)\\ \frac{F}{A}=\frac{y\Delta L}{L}-a\\ F=\frac{Ay\Delta L}{L}-\left(2\right)\end{array}$

$\begin{array}{c}\text{from eqn}1\&2\\ \frac{m{v}^2}{R}=\frac{Ay\Delta c}{L}\\ v=\sqrt{\frac{ARy\Delta L}{mL}}\text{-}\left(3\right)\end{array}$

$\begin{array}{c}T=2\frac{\pi r}{V}-\left(4\right)\\ T=\frac{2\pi M}{\sqrt{\frac{Ary\Delta L}{mL}}}\end{array}$

$\begin{array}{cc}\text{initialy}\Delta L& =2a-a=a\\ \text{final}\Delta L& =3a-a=2a\\ T_1& =\sqrt{\frac{2a}{a}}\\ T_d& =\sqrt{\frac{3a}{20}}\\ & \frac{T}{T_{\varphi }}=\frac{2}{\sqrt{3}}\\ & T_{\varphi }=\frac{\sqrt{3}}{2}T\end{array}$