A solid body starts rotating about a stationary axis with an angular acceleration = at. How soon after the beginning of rotation will the total acceleration vector of an arbitrary point of the body form an angle = 60∘ with its velocity vector?
Total acceleration making 600 with the velocity vector is as good as it making 600 with tangential acceleration since the direction of both v & at is same.
Now from the figure we know
Tan 60o = aNat = √3 .......(i)
A = a x t
Tangential acceleration at = rα = rat ........(ii)
Normal acceleration aN=V2R
V=rω
⇒ aN = rω2
So we need ω or v
Hmmm................... (we cannot use equations of motion as you know α is variable)
α=dωdt=at
⇒∫ω0dω=∫t0at.dt
⇒ω=at22
⇒aN=ra2t44
Now substituting (ii) and (iii) in (i) we get
ra2t44rat=√3
at34=√3
t=(4√3a)13