1. Tardigrade
  2. Question
  3. Physics
  4. A particle is projected horizontally with velocity V0=√2 g a along the smooth inside surface of a fixed hollow hemisphere of inner radius ' a ' at the level of the centre ' O '. The subsequent motion of the particle is confined between the horizontal planes one through the centre and the other at a depth h. Find the value of h :

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Q. A particle is projected horizontally with velocity $V_{0}=\sqrt{2 g a}$ along the smooth inside surface of a fixed hollow hemisphere of inner radius ' $a$ ' at the level of the centre ' $O$ '. The subsequent motion of the particle is confined between the horizontal planes one through the centre and the other at a depth $h$. Find the value of $h$ :

System of Particles and Rotational Motion

Solution:

By conserving angular momentum about vertical axis $O P$
$m v_{0} a=m v \sqrt{a^{2}-h^{2}}\,\,\,...(1)$
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By conserving mechanical energy
$\frac{1}{2} m v^{2}=\frac{1}{2} m v_{0}^{2}+m g h\,\,\,...(2)$
From equation (1) and (2) we get $h=\frac{a(\sqrt{5}-1)}{2}$