A ball is held at a fixed height from a smooth incline plane 'B' of inclination φ = 6 0 ∘ . Now, a smooth platform ' P ' of variable length is placed below the ball and the ball is

Question

A ball is held at a fixed height from a smooth incline plane 'B' of inclination φ = 6 0 ∘ . Now, a smooth platform ' P ' of variable length is placed below the ball and the ball is released from rest. The ball slides on the platform and touches the incline ' B ' in the shortest time? Find the angle of platform ' P ' with vertical (in degrees).

TestHub · Accepted Answer

Ball's acceleration along the platform: $a = g sin (9 0^{\circ} - θ) = g cos θ$ .

From the geometry, platform length $ℓ = \frac{h}{c o s ( ϕ - θ )}$ with $ϕ = 6 0^{\circ}$ .

$t = \frac{2 ℓ}{g c o s θ} = \frac{2 h c o s 6 0 ^{\circ}}{g c o s θ c o s ( 6 0 ^{\circ} - θ )}$ .

Minimising $t$ over $θ$ gives $θ = 3 0^{\circ}$ - (C)

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