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When we see a rope attached to pulley moving down as the wheel of the pulley rotates , then how do we will know that the particles of the rope touching the

wheel are having a linear velocity or an angular velocity ?



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The particles on the rope not touching the pulley have a linear velocity
       = velocity of the moving rope = v

the particles on the rope touching the pulley also have the same linear velocity.  The reason is that all particles on the rope are moving at the same velocity at any instant of time.  The rope is tight and not loose.  The rope has a uniform tension all along the length.  The rope is  mass less, or its mass can be ignored.

Hence, the angular velocity of particles on the rope touching the pulley =
     = linear velocity of particles on rope / radius of pulley

angular velocity of the particles on pulley = same as above,  as there is no slipping of rope on the pulley.

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