A solid disc and a ring, both of radius 10 cm are placed on a horizontal table simultaneously, with initial angular speed equal to `10 π" rad s"^(-1)`. Which of the two will start to roll earlier ? The co-efficient of kinetic friction is `µ_(k) = 0.2`.

Explanation

Radii of the ring and the disc `R=10cm=0.1m`
coefficient kinetic friction `mu_(K)=0.2`
moment of inertia for ring `I=MR^(2)`
moment of inertia for disc `I=(MR^(2))/(2)`
initial angular velocity `omega_(0)=10pi" rads"^(-1)`
`f=ma`
`mu_(K)N=ma" "[f=mu_(K)N]`
`mu_(K)mg=ma" "[because N=mg]`
`therefore a-mu_(K)g" "....(1)`
and torque `tau=-Ialpha`
`therefore Ialpha=-fR [because tau=fR]`
`=-maR`
`=-mu_(K)NR`
`therefore Ialpha=-mu_(K)mgR" "....(2)`
Now in equation of motion `v=u+at,u=0`
`therefore v=at`
`therefore a=(v)/(t)" "....(3)`
From eqn. (1) and (3)
`mu_(K)g=(v)/(t)`
`therefore v=mu_(K)"gt rarr" for ring .....(4)"`
and `v=mu_(K)"gt. "rarr" for disc ....(5)"`
From eqn. (2)
`alpha=-(mu_(K)mgR)/(I)`
From eqn. (2)
`alpha=-(mu_(K)mgR)/(I)`
`alpha=-(mu_(K)mgR)/(mR^(2))rarr` for ring
`=-(mu_(K)g)/(R )`
and `alpha=-(mu_(K)mgR)/(mR^(2))rarr` for ring
`=-(2mu_(K)g^(2))/(R )`
Now in rotational equation `omega=omega_(0)+alphat,`
`omega=omega_(0)-(mu_(K)"gt")/(R )rarr` for ring and
`omega=omega_(0)-(2mu_(K)"gt".)/(R )rarr` for disc
Now `v=Romegararr` condition for rolling of a rigid body
`therefore omega=omega_(0)-(mu_(K)"gt")/(R )`
`therefore (v)/(R)=omega_(0)-(mu_(K)"gt")/(R)[because omega=(v)/(R)]`
`mu_(K)"gt"=R[omega_(0)-(mu_(K)"gt")/(R)][because" From eqn. (4)"]`
`mu_(K)"gt"+mu_(K)"gt = "Romega_(0)`
`2mu_(K)"gt"=Romega_(0)`
`therefore t=(Romega_(0))/(2mu_(K)g)`
`therefore t=(0.1xx10pi)/(2xx0.2xx9.8)`
`therefore t=0.8s rarr` for ring
Now `omega=omega_(0)-(2mu_(K)"gt")/(R )`
`therefore (v)/(R)=omega_(0)-(2mu_(K)"gt")/(R)`
`therefore mu_(K)"gt".=R[omega_(0)-2mu_(K)"gt". [because" from eqn. (5)"]`
`therefore mu_(K)"gt".+2mu_(K)"gt".=Romega_(0)`
`therefore 3mu_(K)"gt".=Romega_(0)`
`therefore t.=(Romega_(0))/(3mu_(K)g)`
`=(0.1xx10pi)/(3xx0.2xx9.8)`
`=0.53s rarr` for disc
`therefore` Since, `t.lt t`, the disc will start rolling before the ring.