A wheel rotates around a stationary axis so that the rotation angle `theta` varies with time as `theta=at^(2)` where `a=0.2rad//s^(2)`. Find the magnitude of net acceleration of the point A at the rim at the moment `t=2.5s` if the linear velocity of the point A at this moment is `v=0.65m//s`.

Explanation

Differentiating `varphi(t)` with respect to time
`(dvarphi)/(dt)=omega_z=2at` (1)
For fixed axis rotation, the speed of the point A:
`v=omegaR=2a tR` or `R=(v)/(2at)` (2)
Differenciating with respect to time
`w_t=(dv)/(dt)=2aR=v/t`, (using 1)
But `w_t=v^2/R=(v^2)/(v//2at)=2atv` (using 2)
So, `w=sqrt(w_t^2+w_n^2)=sqrt((v//t)^2+(2atv)^2)`
`=v/tsqrt(1+4a^2t^4)`