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 · Accepted Answer

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              =  0Which 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

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

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