A hemispherical bowl of radius R is set rotating about its axis of symmetry. A small body kept in the bowl rotates with the bowl without slipping on its surface. If radius through the body makes with the axis an angle θ. θ and assuming that the surface of the bowl is smooth, the angular velocity with which bowl is rotating is given by

Question

A hemispherical bowl of radius R is set rotating about its axis of symmetry. A small body kept in the bowl rotates with the bowl without slipping on its surface. If radius through the body makes with the axis an angle θ. θ and assuming that the surface of the bowl is smooth, the angular velocity with which bowl is rotating is given by

Infinity Learn · Accepted Answer

The different forces on the small body are shown in fig.From figure $$sin$$⁡θ$$=(r/R)$$……………..(i)Now$$, Ncos⁡θ$$=mg$$ ……………………$$(ii)and$$ Nsin⁡θ$$=mr$$ω$$2$$…………………….$$(iii)Dividing$$ eq. $$(iii) by eq. (ii), we gettan⁡θ$$=r$$ω$$2g=(Rsin$$⁡θ$$)$$ω$$2gor$$ $$sin$$⁡θ$$cos$$⁡θ$$=(Rsin$$⁡θ$$)$$ω$$2gor$$ ω$$2=gRcos$$⁡θ  or  ω$$=gRcos$$⁡θ

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