Question

A Toy cart attached to the end of an unstretched string of length a, when received moves on a smooth horizontal table in a circle of radius $2$s 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'. If Hook's law holds then (Assume no friction)

• 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

Answer 1

The correct option is B T' = $\left(\frac{\sqrt{3}}{2}\right)$T

$F=k.\Delta l$

Case 1:

$\Delta l=a$

$F=m{\omega }^{2}r$

$k.a=m{\omega }^2.2a$

$\omega =\sqrt{\frac{k}{2m}}$

Case2:

$\Delta l=2a$

$F=m{\omega }_1^2{r}_1$

$k.2a=m{\omega }_1^2.3a$

${\omega }_1=\sqrt{\frac{2k}{3m}}$

$\frac{T^{\prime }}{T}=\frac{2\pi }{{\omega }_1}\frac{\omega }{2\pi }=\frac{\omega }{{\omega }_1}=\frac{\sqrt{\frac{k}{2m}}}{\sqrt{\frac{2k}{3m}}}=\frac{\sqrt{3}}{2}$

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