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Correct me if I am wrong -
(a) While doing circular motion car is doing pure rotational motion as it rotates around a fixed axis [i am talking about motion of real car which turns its position]

(b) As it is moving friction is only force helping it to do so i.e. friction provide centripetal force.

(c) As being only force - if this force is applied on centre of mass of car it must give me translational component of motion - which obivously not zero.

So Why is circular motion is pure rotational?

Whole Question is based on the following statement of book

We can obtain the translational component of their motion by taking the mass of the whole system to be concentrated at the centre of mass and all the external forces on the system to be acting at the centre of mass.
[I think this statment is wrong]

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I think your problem is that you are trying to make an unnecessary (and unhelpful) distinction between circular motion and translational motion.

When motion is divided between rotational and translational, the motion of the centre of mass is taken to be the translational and any motion of other parts of the object about the CM is rotational. In this case the CM of the car is moving, so there is translational motion. Also the direction in which the car is pointing rotates, so there is rotational motion.

Circular motion is not necessarily the same as rotational motion.

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