Sliding bead

Classical Mechanics Level 3

A bead is released from a certain point and slides down a frictionless wire that connects this point to another point, as shown in the figure above. What shape should the wire take so that the bead reaches the endpoint in the shortest possible time?

Hyperbolic spiral Straight line Cycloid Parabolic

Beakal Tiliksew by Beakal Tiliksew , 24, Ethiopia

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4 solutions

Beakal Tiliksew
Apr 10, 2014

This is called the brachistochrone .

13 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice gif! Did you make it yourself? If so, how did you do it?

Calvin Lin Staff - 7 years, 2 months ago

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Actually i found it online

Beakal Tiliksew - 7 years, 2 months ago

In all paths energy stored in bed will be same but when on releasing bed it will took same time via all paths... Am i right ??? :[

lalit bonde - 7 years, 1 month ago

Thanks

Mardokay Mosazghi - 7 years, 2 months ago

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Ur welcome:)

Beakal Tiliksew - 7 years, 2 months ago

ok

Bahaa Salah - 7 years, 1 month ago

Why not a Hyperbolic Spiral?

Maham Zaidi - 7 years, 2 months ago

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Any reason why it should be a hyperbolic spiral?

Beakal Tiliksew - 7 years, 2 months ago

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Actually I was imagining it, how the speed would go and how less time it will take. The spiral seemed like a good enough answer.

Maham Zaidi - 7 years, 2 months ago

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@Maham Zaidi – It can't be a hyperbolic spiral as taking the origin as the two perpendicular lines , this becomes a rectangular hyperbola, which only tends to 0 , but here it is clearly becoming 0 for two values

Calvin David - 7 years, 1 month ago

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@Calvin David – Got it (:

Maham Zaidi - 7 years, 1 month ago

@Maham Zaidi – The math don't lie.

Beakal Tiliksew - 7 years, 1 month ago

I dont think it will be a cycloid ghis time as the brachistochrone is a cycloid when initial and final points are at the same level.......

Rishabh Mishra - 6 years, 3 months ago
Steven Zheng
Jul 15, 2014

I'm not sure why this is level 3, but I like the gif. I also like cycloids, so I like this question.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

it should be level 2 or level 4/5 ? ccording to you ...??

A Former Brilliant Member - 4 years, 4 months ago
Balaji Dodda
Apr 21, 2014

Calculus of Variations: extremize the integral : sqrt(1+(dy/dx)2)/sqrt(19.6 y). This will give you the minimum time. the numerator is the distance in infinitesimal calculus and denominator is the velocity as the particle moves along the trajectory. velocity sqrt(19.6 y) can be obtained by conservation of energy. Now solve this using Lagrange's equation popular in classical mechanics or calculus of variations.

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It is similar to the brachistochrone problem, just that in the brachistochrone problem the wires are in the shape of an inverted cycloid (a tautochrone), an incline and a cycloid.

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