Newtonian mechanics and capstan

[Kolahal Bhattacharya]

Consider a capstan.A circular body around which a strong rope is wound.The free ends of the rope are pulled with tension Ta and Tb. Ta>>Tb.Co--eff. of friction is mu.theta is the total angle subtended by the attached portion of the rope on the centre of the drum.Prove that Ta=Tb[exp(-mu theta)]

 

[Chi Meson]

https://www.physicsforums.com/showthread.php?t=94379

Not only that, but the description is not very clear. Got a drawing?

 

[Kolahal Bhattacharya]

re Newtonian mechanics

I cannot give the picture here.Pleaase refer to Kleppner and Kolenkow 2.24 for this problem.My calclation gives Ta= Tb+(mu)N, where N has the same value as the centripetal force.according to the answer, N should be equal to (Ta-Ta exp(-mu theta)). I could not get it

 

[J Hann]

You need to derive the following equations:

(T + dT) cos dtheta - T - mu N ds = 0
(T + dT) sin dtheta - N ds = 0 where N is the normal force / unit length
Use the fact that dtheta is small and eliminate N
This is called Tension on a Cord Wrapped around a Rough Post