A solid disc and a string, both of radius 0.10m are placed on a horizontal table simultaneously, with the initial angular speed equal to `10pi rad^(-1)`. Which of the two will start to roll earlier ? The coefficient of kinetic friction is `mu_k=0.2`

Explanation

Frictional torque`=Iprop`
but `mu_kmgR=-Iprop` ( since the motion is retarding)
`:.prop=-(mu_kmgR)/(I)`
for a disc `I=(mR^2)/2` and for a ring I=`MR^2" "prop=(-2mu_kmgR)/(mR^2)=-(2mu_kg)/R` for a disc
and `prop=(-mu_kmgR)/(mR^2)=(-mu_kg)/(R)` for a ring. Applying `omega=omega_0+propt` for a ring
We write `omega=omega_0-(mu_kg t_1)/(R)`
but `omega=v/R=(mu_kg t_1)/R` where `a=mu_kg`
`(mu_kg t_1)/R+(mu_kg t_1)/R=omega_0`
i.e `2(mu_kg t_1)/R=omega_0" or "t_1=(omega_0R)/(2mu_1g)`
Similarly for a disk `(mu_kg t_2)/R=omega_0-(2mu_g)/Rg t_2`
`omega_0=(3mug t_2)/R`
and `t_2=(omega_1R)/(3mu_kg)`
we note that `t_2 lt t_1`. Hence the disc rolls down earlier than the ring.