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belt, the magnitude of the frictional force dF = µ dN . This force
opposes the sliding motion of the belt, and so it will increase the
magnitude of the tensile force acting in the belt by dT. Applying the
two force equations of equilibrium, we have

T
Σ F = 0; cos   dθ   + µ dN − (T + dT )cos   dθ   = 0
x
 2   2 
 dθ   dθ 
Σ F = 0; dN − (T + dT )sin   − T sin   = 0
y
 2   2 
Since dθ is of infinitesimal size, sin(dθ / 2) = dθ / 2 and
cos(dθ / 2) 1= . Also, the product of the two infinitesimals dT and
dθ /2 may be neglected when compared to infinitesimals of the first
order. As a result, these two equations become
µ dN = dT
and

dN = Tdθ
Eliminating dN yields
dT = µθ
d
T
Integrating this equation between all the points of contact that
T
the belt makes with the drum, and noting that T = at θ = 0 and
1
T = T at θ = β , yields
2
∫ 1 T 2 T dT = µ ∫ 0 β dθ
T
T
ln 2 = µβ
T 1
Solving for T , we obtain
2
µβ
T = Te 1-53
1
2
where T , T = belt tensions T ; opposes the direction of motion (or
2
1
1
impending motion) of the belt measured relative to the surface, while
T acts in the direction of the relative belt motion (or impending
2
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