Lab 8.Cable in Contact with a Rough Surface.Here we consider the mathematics of a rope or cable passing over a rough surface. Youmay have already come across the phenomenon of wrapping a rope around a tree or a post,for example, and producing a situation in which huge loads can be held. You may haveseen tree cutters use this to lower branches.Consider the situation as depicted in the sketch.We will now balance the forces acting on a small piece of the cable ∆s.First we balance the forces in the tangential direction:TFs T T+=+∆∆∆( )cos( )θ.If we use the tangent line approximation for cos ( )∆θ, which is simply cos( )∆θ≈ 1, wehave,TFs T T+≈+∆∆().Thus on letting ∆s→0, we derive,(i) dTdsF= .Secondly we balance the forces in the direction perpendicular to the tangent, that is to sayalong the normal,Ns T T∆∆∆θ=+( )sin( ).If we use the tangent line approximation for sin ( )∆θ, which is sin( )∆∆θθ≈, we have,∆θ∆θT , FT+∆TNNs T T∆∆∆θ≈+(),and so if we let ∆s→0, we derive,(ii)NdsdTθ= .However if the rope is on the point of slipping then(iii) FN=µ ,where the symbol µ (pronounced mu ) is the coefficient of friction.The larger this coefficient is the rougher the surface.Now work the following problems.(a) By eliminating F and N from the equations (i), (ii) and (iii) show that(iv)dTdTθµ= .This is an old friend, it shows that T increases exponentially with θ.(b) Now let’s work a problem of a rope wrapped around a post.Determine how much initial tension is needed, that is to say the tension in the rope when itfirst meets the post, in order for it to support a one ton weight if the rope is wrappedaround the post two times. Let the coefficient of friction be 0.5.(c) Recall that we said that equation (iii) holds only if the rope is on the point of slipping.In general the friction coefficient µ will be a function. In our problem let’s suppose thatthe friction depends on the tension in such a way thatµ= ≤ ≤=<TTT/, ,/, .20 0 1012 10forforRedo problem (b) with this function replacing the constant value we assumed