Question

A Toy cart attached to the end of an unstretched string of length $a$, when revolved moves on a smooth horizontal table in a circle of radius $2a$ with a time period $T$. Now the toy cart is speeded up until it moves in a circle of radius $3a$ with a period $T^{\prime }$. If Hook's law holds then (Assume no friction):

• A. $T^{\prime }=\sqrt{\frac{3}{2}}$ T
• B. $T^{\prime }=\left(\frac{\sqrt{3}}{2}\right)$ T
• C. $T^{\prime }=\left(\frac{3}{2}\right)$ T
• D. $T^{\prime }=T$

Answer 1

Answer: (B)

$T^{\prime }=\left(\frac{\sqrt{3}}{2}\right)$ T

The extension in the string in first case $=(2a-a)=a$
Now Hooke's law is $F=kx$
$\therefore m\left(2a\right){\omega }^2=k.a$
$m\left(2a\right)(\frac{2\pi }{T}{)}^2=ka$ .....................(1)
In second case
$m\left(3a\right)(\frac{2\pi }{T}{)}^2=k(3a-a)$ .......................(2)
Dividing (1) and (2) and solving,we get
$T^{\prime }=\frac{\sqrt{3}}{2}T$
hence, option B is correct.