Question

A toy car travels in a horizontal circle of radius $2a$, kept on the track by a radial elastic string of unstretched length $a$. The period of rotation is $T$. Now the car is speeded up until it is moving in a circle of radius $3a$. Assuming that the string obeys Hookes law then the new period will be

• A. $\sqrt{\frac{4}{3}} T$
• B. $\frac{3^{2}}{4^{2}} T$
• C. $\frac{\sqrt{3}}{2} T$
• D. $ \frac{3}{4}T $

Answer 1

Answer: (C)

For a body executing circular motion, centripetal force is given by

$F=m r \omega^{2}=m r\left(\frac{2 \pi}{T}\right)^{2}$ ...(i)
If $k$ is force constant of string, then elastic force
$F=k a$ ...(ii)
From Eqs. (i) and (ii), we get
$k a=m(2 a)\left(\frac{2 \pi}{T}\right)^{2}$ ...(iii)
In second case New length of string = new radius of circle $=3a$
Stretching of string $=3 a-a=2 a$
Hence, elastic force $=k \cdot 2 a$
So, $k \cdot 2 a=m(3 a)\left(\frac{2 \pi}{T'}\right)^{2}$ ...(iv)
Dividing Eq. (iv) by Eq. (iii), we get
$2 =\frac{3}{2}\left(\frac{T}{T'}\right)^{2}$
$\Rightarrow \left(\frac{T}{T'}\right)^{2} =\frac{4}{3}$
$\Rightarrow T'=\frac{\sqrt{3}}{2} T$