saucletbh
2022-09-26
Answered

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

Answered 2022-09-27
Author has **13** answers

The acceleration vector in polar coordinates for a general non-uniform 2D planar motion can be described as:

$\mathbf{a}=[\frac{{d}^{2}r}{d{t}^{2}}-r{\left(\frac{d\theta}{dt}\right)}^{2}]\hat{r}+[2\frac{dr}{dt}\frac{d\theta}{dt}+r\frac{{d}^{2}\theta}{d{t}^{2}}]\hat{\theta}$

Now, for non-uniform circular motion, the distance r from axis of motion is fixed:

$\frac{dr}{dt}=0\Rightarrow \frac{{d}^{2}r}{d{t}^{2}}=0$

Which gives the acceleration vector as:

$\mathbf{a}=-r{\left(\frac{d\theta}{dt}\right)}^{2}\hat{r}+r\frac{{d}^{2}\theta}{d{t}^{2}}\hat{\theta}$

The $\hat{r}$ 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 $r=r(\theta )$

You can get different spirals depending on the exact form of $r(\theta )$, the equations of which can be found

$\mathbf{a}=[\frac{{d}^{2}r}{d{t}^{2}}-r{\left(\frac{d\theta}{dt}\right)}^{2}]\hat{r}+[2\frac{dr}{dt}\frac{d\theta}{dt}+r\frac{{d}^{2}\theta}{d{t}^{2}}]\hat{\theta}$

Now, for non-uniform circular motion, the distance r from axis of motion is fixed:

$\frac{dr}{dt}=0\Rightarrow \frac{{d}^{2}r}{d{t}^{2}}=0$

Which gives the acceleration vector as:

$\mathbf{a}=-r{\left(\frac{d\theta}{dt}\right)}^{2}\hat{r}+r\frac{{d}^{2}\theta}{d{t}^{2}}\hat{\theta}$

The $\hat{r}$ 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 $r=r(\theta )$

You can get different spirals depending on the exact form of $r(\theta )$, the equations of which can be found

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