A block of mass `m` is kept on a horizontal ruler . The friction coefficient between the ruler and the block is `m` . The ruler is fixed at one end the block is at a distance L from the fixed end . The ruler is rotated about the fixed end in the horizontal plane through the fixed end
a. What can the maximum angular speed be for which the block does not slip?
b. If the angular speed of the ruler is uniform increase from zero at an angular acceleration a at angular speed will the block slip?

Explanation

a. If the block does not slip `f le mu N`
`m omega^(2)L le mu mg rArr omega le ((mu g)/(L))^(1//2)`
b. If is the case of non uniform circular motion
`f_("tangential")= m(dv)/(dt) = m(d(omega L))/(dt) = mL(d omega)/(dt) = maL`
`F_("tangential") = m omega^(2)L`....(i)
`F_("radial") = m omega^(2)L`....(ii)
Net force `F_("net") = sqrt((maL)^(2) + (m omega^(2)L)^(2))`
Friction force `f = F_("net") rArr f = sqrt((maL)^(2)+ (m omega^(2)L)^(2))`
`f le mu N rArr sqrt((maL)^(2) + (m omega^(2)L)^(2)) le mu mg `
`m^(2)alpha^(2)L^(2) + m^(2)omega^(4)L^(2) le mu^(2) m^(2) g^(2)`
` rArr omega le [((mu g)/(L))^(2) - alpha^(2)]^(1//4)`