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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. |
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Answer» Solution :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. |
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