Question

A loop of a string of mass per unit length $\mu$ and radius R is rotated about an angle passing through centre perpendicular to the plane with n angular velocity $\omega$. A small disturbance is created in the loop having the same sense of rotation. The liner speed of the disturbance for a stationary observer is

• A. $\omega$R
• B. 2$\omega$R
• C. 3$\omega$R
• D. zero

Answer 1

The correct option is B 2$\omega$R

$2Tsin\frac{d\theta }{2}=dm{\omega }^{2}R$

$2T\left(\frac{d\theta }{2}\right)=\lambda Rd\theta {\omega }^{2}R$

$\frac{T}{\lambda }={\omega }^2{R}^2$

Therefore,

$\sqrt{\frac{T}{\lambda }}=\omega R$

$\sqrt{\frac{T}{\lambda }}=V_{pulse\&string}=V_{pulse}-V_{string}⟹\omega R=V_P-V_S$

Then,

$V_P=\omega R+V_S=\omega R+\omega R=2\omega R$