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

A solid disc and a string, both of radius 0.10m are placed on a horizo

1.	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
Answer» Solution :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.