1.

A long horizontal plank of mass m is lying on a smooth surface. A solid sphere of same mass m and radius 'r' is spinned about its own axis with angular velocity omega_(0) and gently placed on the plank. The coefficient of friction between the plank and the sphere is mu. After some time the pure rolling of the sphere on the plank will start. Find the displacement of the plank till the sphere starts pure rolling

Answer»

`(omega_(0)^(2)r^(2))/(81mug)`
`(2)/(27)(omega_(0)^(2)r^(2))/(mug)`
`(4)/(81)(omega_(0)^(2)r^(2))/(mug)`
`(2)/(81)(omega_(0)^(2)r^(2))/(81mug)`

SOLUTION :`V_(1)=(2)/(sqrt(5))sqrt(5gR)=2sqrt(gR)`
`V_(2)^(2)=V_(1)^(2)-2g(R+Rsintheta)=2gR-2gRsintheta`
N + `mgsintheta=(mV_(2)^(2))/(R )`
Putting N = 0 implies `mgsintheta = m2g(1-sintheta)`
` implies theta=sin^(-1)((2)/(3))`
`V^(2)=V_(1)^(2)-2gRimpliesV^(2)=4gR-2gR`
`V=sqrt(2gR)`
`a_(c )=(V^(2))/(R )=2gimpliesa_(t)=g`
`TAN alpha=(a_(t))/(a_(c ))impliesalpha=tan^(-1)((1)/(2))`
Maximum contact force is at A.


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