Alternative approach to analyzing a massless string

[Amin2014]

Consider a massless string which can rotate about a fixed pulley (first picture). The coefficient of static friction is μ. Assuming that the motion is impending, the goal is to find the equation that describes the variation in tension of the string.
( T₂/T₁ = e^μΦ where Φ is the subtended angle.)

The usual method of solving this problem involves writing Newton's law for an infinitesimal element of the string and then integrating. In the second picture I've provided an extract from a different problem. Why can't we apply the solution provided in the second picture to the original problem stated above?

 

[Amin2014]

Here's my own solution:
N Cos β =μ N Sinβ therefore tan β = 1/μ
Writing Σ M = 0 about the point of application of R and taking the radius of the pulley to be r we have :
T₂ ( r - rCos β) = T₁ (r + r Cosβ)
dividing by r and rearranging:
T₂ = T₁ (1 + Cosβ) / ( 1 - Cos β)
Which yields a different relation between T₁ and T₂ than the method of integration.

 

[Amin2014]

Consider a massless string which can rotate about a fixed pulley (first picture). The coefficient of static friction is μ. Assuming that the motion is impending, the goal is to find the equation that describes the variation in tension of the string.
( T₂/T₁ = e^μΦ where Φ is the subtended angle.)

The usual method of solving this problem involves writing Newton's law for an infinitesimal element of the string and then integrating. In the second picture I've provided an extract from a different problem. Why can't we apply the solution provided in the second picture to the original problem stated above?

Click on "alternative Solution.png", for better image quality.