How much Rope can be there?

Classical Mechanics Level 4

A rope rests on two platforms that are both inclined at an angle θ \theta (which you are free to pick), as shown above. The rope has uniform mass density, and its coefficient of friction with platform is 1. The system has left-right symmetry. What is the largest possible fraction of the rope that does not touch the platforms?

2 − 1 2 + 1 \frac { \sqrt { 2 } -1 }{ \sqrt { 2 } +1 } 2 + 1 2 − 1 \frac { \sqrt { 2 } +1 }{ \sqrt { 2 } -1 } 2 2 + 1 \frac { \sqrt { 2 } }{ \sqrt { 2 } +1 } 2 + 1 2 \frac { \sqrt { 2 } +1 }{ \sqrt { 2 } }

Rohit Gupta by Rohit Gupta , 34, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Ashwin Gopal
May 23, 2015

This is a famous equilibrium problem. One of the approach is by considering the rope as 3 blocks having mass kx,ky and kx where k is mass per unit length and x is length lying on inclined plane and y be the length of hanging part. Now draw FBD and solve for equilibrium condition. U will get fraction as f = ( F ( θ ) / [ F ( θ ) + 1 ] ) w h e r e F ( θ ) = ( sin ⁡ θ × cos ⁡ θ − sin ⁡ θ 2 ) f=(F(\theta)/[F(\theta)+1]) where F(\theta) = (\sin \theta\times \cos \theta - \sin \theta ^{2}) Now applying calculus, we get fraction.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Yups its a famous and innovative kinda problem..!! The first timers generally feel difficulty with it..!!

Rohit Gupta - 6 years ago

Log in to reply

There is very good solution to this problem in the Internet. (in PDF form)

Ashwin Gopal - 6 years ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...