Q. Toy cart tied to the end of an unstretched string of length $a$, when revolved 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 Hook’s law holds then
A) $T^{'} = \sqrt{\frac{3}{2}}\ T$
B) ${T^{'}}^{\ } = \left( \frac{\sqrt{3}}{2} \right)T$
C) $T^{'} = \left( \frac{3}{2} \right)T$
D) $T^{'} = T$
Explanation
(a) Extension is first case
$\mathrm{\Delta}l_{1} = 2a - a = a$
From Hooke’s law
$F = k\mathrm{\Delta}l_{1}$
$mr\omega^{2} = ka$
$m(2a)\omega^{2} = ka$
$2ma\left( \frac{2\pi}{T} \right)^{2} = ka$
In second case,
$m(3a)\left( \frac{2\pi}{T^{'}} \right)^{2} = k(3a - 2a)$
On dividing Eq. (ii) by Eq. (i)
$T^{'} = \sqrt{\frac{3}{2}}T$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.