A-point-P-is-located-above-an-inclined-plane-It-is-possible-to-reach-the-plane-by-sliding-under-gravity-down-a-straight-frictionless-wire-joining-to-some-point-P-on-the-plane-How-should-P-be-cho

Question Number 18262 by Tinkutara last updated on 17/Jul/17

A point P is located above an inclined

plane . It is possible to reach the plane

by sliding under gravity down a straight

frictionless wire joining to some point

P^{'} on the plane . How should P^{'} be

chosen so as to minimize the time

taken ?

Commented by Tinkutara last updated on 17/Jul/17

Commented by ajfour last updated on 18/Jul/17

Answered by ajfour last updated on 18/Jul/17

Perpendicular from P to incline

be D .

let time taken to reach P^{'} be t .

length of wire = Dsec (\frac{π}{2} - α - θ)

= \frac{D}{\sin (θ + α)}

acceleration along wire = gsin θ

\frac{D}{\sin (θ + α)} = \frac{{gt}^{2} \sin θ}{2}

t^{2} = \frac{2 D}{gsin θ \sin (θ + α)} > 0

= \frac{4 D}{\cos α - \cos (2 θ + α)}

For t^{2} to be minimum,

denominator is a maximum .

that is \cos (2 θ - α) = - 1

2 θ - α = π

θ = \frac{π}{2} - \frac{α}{2} .

wire is then the angular bisector

of ⊥ from P to incline plane and

vertical line from P to incline plane .

Commented by Tinkutara last updated on 19/Jul/17

Thanks Sir!