A parabolic bowl with its bottom at origin has the shape $y = \frac{x^{2}}{20}$ . Here $x$ and $y$ are in metres. The maximum height at which a small mass $m$ can be placed on the bowl without slipping (coefficient of static friction is $0.5$ ) is

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Slope, d y d x = x 10 or tan ⁡ θ = x 10 ... (i) Equilibrium of mass in horizontal direction gives the equation, μ N cos ⁡ θ = N sin ⁡ θ or tan ⁡ θ = μ = 1 2 ... (ii) From Equations. (i) and (ii) x 10 = 1 2 or x = 5 m ∴ y = x 2 20 = 25 20 = 1.25 m

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