Question

A circular disc of mass M and radius R is connected to spring and placed on rough surface. The friction is sufficient for pure rolling. When the disc is slightly displaced and released, it executes simple harmonic motion.

Let $\mu$ be the coefficient of friction between the disc and ground, the amplitude A of oscillations for which no slipping occurs is

• A. $0<A<\frac{4\mu Mg}{K}$
• B. $A>\frac{3\mu Mg}{K}$
• C. amplitude A must be less than $\frac{\mu Mg}{K}$
• D. $0<A<\frac{3\mu Mg}{K}$

Answer 1

The correct option is D $0<A<\frac{3\mu Mg}{K}$

$F-f=mafR=\frac{m{R}^2\alpha }{2}f=\frac{ma}{2}F=\frac{3ma}{2}F=3ff=\frac{F}{3}f=\frac{Kx}{3}X=acos\omega tf=\frac{KA}{3}cos\omega t\mu mg=\frac{KA}{3}A=\frac{3\mu mg}{K}$
$\therefore 0<A<\frac{3\mu mg}{K};$ for no slipping
$a=\frac{-2F}{3m}=\frac{-2Kx}{3m}T=2\pi \sqrt{\frac{3m}{2k}}{W}_f=0$