Derve the relation between tangential acceleration and angular acceleration.

Explanation

Consider an object moving along a circle of radius r. In a time `Deltat`, the object travels in an are distance As as shown in figure. The corresponding angle subtended is `Deltatheta`. The `Deltas` can be written in terms of `Deltatheta` as,

`Deltas=rDeltatheta" "......(1)`
In a time `Deltat`, we have
`(Deltas)/(Deltat)=t""(Deltatheta)/(Deltat)" "......(2)`
In the limit `Deltat"to0`, the above equation becomes
`(ds)/(dt)=romega" ".......(3)`
Here `(ds)/(dt)` is linear speed (5) which is tangential to the circle and ois angular speed. So equation (3) becomes
`v_(r)=romega" "......(4)`
which gives the relation between linear speed and angular speed.
Eq. (4) is true only for circular motion. In general the relation between linear and angular velocity is given by
`vecv=vecomegaxxvecr" ".......(5)`
For circular motion eq. (5) reduces to eq. (4) since `vecomegaandvecr` are perpendicular to each other. Differentiating the eq. (4) with respect to time, we get (since r is constant)
`(dv)/(dt)=(rdomega)/(dt)=ralpha`
Here `(dv)/(dt)` is the tangential acceleration and is denoted as `a_(t)=(domega)/(dt)` is the angular acceleration `alpha`. Then eq. (5) becomes
`a_(t)=ralpha" "......(6)`.