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A solid sphere of radius R is projected on a rough horizontal surface of coefficient of friction with linear Speed v_(0), at time to 0. It slips for some time, finally starts to roll with out slipping at time t =t_(0) .Find a) The linear velocity of its centre when pure rolling begins. b) The time t_(0) |
Answer» SOLUTION : a) The kinematics equations for the motion of the sphere at time t are `v=v_(0)`-at ,where a =`mug` `omega=alphat`,where `alpha=(mumgR)/((2)/(5)mR^(2))=(5a)/(2R)`,when pure rolling begins `v^(|)=omega^(|)R` (at t=`t_(0)`) `underset((v^(|)/(R)=(5a)/(2R)t_(0)))(V^(|)=V_(0)-at_(0))}implies(7V^(|))/(5)=V_(0)` or `V^(|)=(5V_(0))/(7)implies` REQUIRED linear velocity =`V^(|)=(5V_(0))/(7)` b)`v^(|)=v_(0)mugt_(0)impliesmugt_(0)=v_(0)-v^(|)=(2v_(0))/(7)impliest_(0)=(2v_(0))/(7 mug)` ALITER: The angular momentum of the sphere about the horizontal surface is conserved. `impliesL_("initial")`(at t=`t_(0)`) `impliesMv_(0)R=mv^(|)R+(2)/(5)mR^(2)omega^(I)=mv^(I)R^(2)xx(v^(I))/(R)=(7)/(5)mv^(I)RimpliesV^(I)=(5)/(7)v_(0)` |
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