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For small α we have sin α ≈ tan α = ∂u (0, t). Since both forces should cancel we
∂x
find
T ∂u
u(0, t) − = 0, t > 0,
k ∂x
and
T ∂u
u(0, t) − = 0, t > 0,
k ∂x
These are mixed boundary conditions.

force from spring

.

α

String tension
parallel to beam

Figure 5.7: The balance of forces at one endpoint of the string of Fig. 5.6

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