1.
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\caption{Figure 1}
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A hemispherical bowl, of internal radius \(r\), is fixed with its circular rim upwards and horizontal. A particle \(P\) of mass \(m\) moves on the smooth inner surface of the bowl. The particle moves with constant angular speed in a horizontal circle. The centre of the circle is at a distance \(\frac { 1 } { 2 } r\) vertically below the centre of the bowl, as shown in Figure 1.
The time taken by \(P\) to complete one revolution of its circular path is \(T\).
Show that \(T = \pi \sqrt { \frac { 2 r } { g } }\).