What path should you choose to reach in the shortest time?
The inclined triangle with a angle is fixed on the ground. The peg on the left wall is also fixed at a height . The string is extended from the peg to the red ring which can slide over the inclined plane. Your task is to slide the ring to some position on the inclined triangles hypotenuse such that a small ball can slide from the peg on the left wall to the chosen position on the inclined plane such that the time it takes is minimum .
What is the distance of the chosen point from the peg on the wall?
Assume that the string is infinitely stretchable and can assume any length and that the shape of the string does not bend under the weight of the ball as it slides down the string.
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1 solution
Notice the fact that if you have a vertical disk with grooves starting from the topmost point in all direction, a small particle takes exactly the same time to slide down each one of these grooves! . Moreover, the time increases as radius of the disk increases. Hence, our answer is the shortest distance from the peg to the inclined plane which is ..... obviously ....... h cos θ !