A disc of radius R rotates from rest about a vertical axis with a constant angular acceleration such that a coin placed with a tangenitial between a as shown in figure If the coefficient of static friction between the coin and disc is `mu_(s)` .Find the velocity of the before it start sliding relative as the disc

Explanation

As the coin move in a circle it experiences radial force `F` , and tangential force `F_(t) F_(r)`and `F_(t)` are the components of friction `f_(s)`.
Force equation `F_(r) = ma_(r)`…(i)
Since `(a_(t) = a`: given), `F_(t) = ma_(t) = ma` ...(ii)
`sum F_(y) = N - mg = ma_(r)`....(iii)
Law of static friction `f_(s) le mu_(s) N` ...(iv)
Kinematics , `a_(r) = (v^(2))/(R)`...(v)
Since the disc does not move vertical `a_(y) = 0`
Vector addition of forces
`sqrt(F_(t)^(2) + F_(r)^(2)) le f_(s)`
From Eqs(i) and (v) , we have `F_(r)= (mv^(2))/(R)`
From Eqs(iii) and (iv) , we have `N = mg` substituting `N = mg` in Eq (iv) we have
`f_(s) = mu_(s) mg`
substittating `F_(t) F_(r)` and `f_(s)` we have
`(m^(2)v^(4))/(R^(2)) + m^(2) a^(2) le mu_(s)^(2) m^(2) g^(2)`
`v le sqrt(Rsqrt(mu_(s)^(2)g^(2)-a^(2)))`