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\includegraphics[max width=\textwidth, alt={}, center]{9bc86277-9e6b-41f6-a2c3-94c85e7b1360-4_330_1061_989_267}
The flat surface of a smooth solid hemisphere of radius \(r\) is fixed to a horizontal plane on a planet where the acceleration due to gravity is denoted by \(\gamma\). \(O\) is the centre of the flat surface of the hemisphere.
A particle \(P\) is held at a point on the surface of the hemisphere such that the angle between \(O P\) and the upward vertical through \(O\) is \(\alpha\), where \(\cos \alpha = \frac { 3 } { 4 }\).
\(P\) is then released from rest. \(F\) is the point on the plane where \(P\) first hits the plane (see diagram).
- Find an exact expression for the distance \(O F\).
The acceleration due to gravity on and near the surface of the planet Earth is roughly \(6 \gamma\).
- Explain whether \(O F\) would increase, decrease or remain unchanged if the action were repeated on the planet Earth.
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