A block of mass m is kept on a horizontal ruler. The friction of coefficient between the ruler and the block is `mu`. The ruler is fixed at one end and 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 of the ruler b) if the angular speed of ruler is uniformly increased from zero to an angular acceleration `alpha` , at what 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)`