# How can we say that a body is doing circular motion while doing a non uniform circular motion?

How can we say that a body is doing circular motion while doing a non uniform circular motion if the centripetal force is changing?
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Lorenzo Acosta
The acceleration vector in polar coordinates for a general non-uniform 2D planar motion can be described as:
$\mathbf{a}=\left[\frac{{d}^{2}r}{d{t}^{2}}-r{\left(\frac{d\theta }{dt}\right)}^{2}\right]\stackrel{^}{r}+\left[2\frac{dr}{dt}\frac{d\theta }{dt}+r\frac{{d}^{2}\theta }{d{t}^{2}}\right]\stackrel{^}{\theta }$
Now, for non-uniform circular motion, the distance r from axis of motion is fixed:
$\frac{dr}{dt}=0⇒\frac{{d}^{2}r}{d{t}^{2}}=0$
Which gives the acceleration vector as:
$\mathbf{a}=-r{\left(\frac{d\theta }{dt}\right)}^{2}\stackrel{^}{r}+r\frac{{d}^{2}\theta }{d{t}^{2}}\stackrel{^}{\theta }$
The $\stackrel{^}{r}$ component of acceleration is the centripetal acceleration:
${a}_{r}=-r{\left(\frac{d\theta }{dt}\right)}^{2}=-\frac{{v}^{2}}{r}$
where $v=r\frac{d\theta }{dt}$, is time-dependent. Hence, we can have a time-varying centripetal acceleration while keeping r fixed, which is the essential constraint for circular motion of any kind.
Now, for a spiral motion, r is not fixed and is usually expressed as a function of $\theta$, such that $r=r\left(\theta \right)$
You can get different spirals depending on the exact form of $r\left(\theta \right)$, the equations of which can be found