7 CIRCULAR MOTION 7.5 Non-uniform circular motion
Comparing with Eq. (7.28), recalling that e i θ represents er and i e i θ represents
eθ, we obtain
ar = ̈r − r θ ̇^2 = ̈r − r ω^2 , (7.41)
aθ = r θ ̈^ + 2 ̇r θ ̇^ = r ω ̇ + 2 ̇r ω. (7.42)Thus, we now have expressions for the object’s radial and tangential accelerations
in terms of r and θ. The beauty of this derivation is that the complex analysis
has automatically taken care of the fact that the unit vectors er and eθ change
direction as the object moves.
Let us now consider the commonly occurring special case in which an object
executes a circular orbit at fixed radius, but varying angular velocity. Since the
radius is fixed, it follows that ̇r = ̈r = 0. According to Eqs. (7.3 8 ) and (7.3 9 ), the
radial velocity of the object is zero, and the tangential velocity takes the form
vθ = r ω. (7.43)Note that the above equation is exactly the same as Eq. (7.6)—the only difference
is that we have now proved that this relation holds for non-uniform, as well as
uniform, circular motion. According to Eq. (7.41), the radial acceleration is given
by
ar = −r ω^2. (7.44)The minus sign indicates that this acceleration is directed towards the centre
of the circle. Of course, the above equation is equivalent to Eq. (7.15)—the only
difference is that we have now proved that this relation holds for non-uniform, as
well as uniform, circular motion. Finally, according to Eq. (7.42), the tangential
acceleration takes the form
aθ = r ω ̇. (7.45)The existence of a non-zero tangential acceleration (in the former case) is the
one difference between non-uniform and uniform circular motion (at constant
radius).