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

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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Answers (1)

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:
a = [ d 2 r d t 2 r ( d θ d t ) 2 ] r ^ + [ 2 d r d t d θ d t + r d 2 θ d t 2 ] θ ^
Now, for non-uniform circular motion, the distance r from axis of motion is fixed:
d r d t = 0 d 2 r d t 2 = 0
Which gives the acceleration vector as:
a = r ( d θ d t ) 2 r ^ + r d 2 θ d t 2 θ ^
The r ^ component of acceleration is the centripetal acceleration:
a r = r ( d θ d t ) 2 = v 2 r
where v = r d θ d t , 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 θ, such that r = r ( θ )
You can get different spirals depending on the exact form of r ( θ ), the equations of which can be found
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