A particle of mass \( m \) is placed in equilibrium at the top of a fixed rough hemisphere of ra....

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A particle of mass \( m \) is placed in equilibrium at the top of a fixed rough hemisphere of radius \( R \). Now the
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particle is given a gentle push so that it starts sliding on the surface of the hemisphere. It is found that the particle leaves the contact with the surface of the hemisphere at angular position \( \theta \) with the vertical where \( \cos \theta=\frac{3}{5} \). If the work done against friction is \( \frac{x m g R}{20} \), find \( x \).


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