A small sized mass m is attached by a massless string (of length L) to the top of a fixed frictionless solid cone whose axis is vertical. The half angle at the vertex of the cone is θ. If the mass m moves around in a horizontal circle at speed v, what is the maximum value of v for which mass stays in contact with the cone ? (g is acceleration due to gravity.)

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At maximum velocity the mass will just loose contact with cone and will behave like free conical pendulum with time period
Hence