Assertion (A): A disc rolls without slipping on a fixed rough horizontal surface. Then there is no point on the disc whose velocity is in vertical direction.
Reason (R): Rolling motion can be taken as combination of translation and rotation. Due to the translational part of motion a velocity (translational component) exist in horizontal direction for any point on the disc rolling on a fixed rough horizontal surface.
Solution:
The velocity of any point on a rolling disc is \( \vec{v} = v_{CM}\hat{i} + (\vec{\omega} \times \vec{r}) \). For pure rolling, \( v_{CM} = \omega R \). If \( v_x = v_{CM} + \omega y = 0 \), this occurs only at the contact point (\( y = -R \)), where \( v_y = -\omega x = 0 \). Thus, no point has a purely vertical velocity. Both A and R are true, and R explains A.
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