2X
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2X L θ T 2 2σds λgds T 1 Figure 1 The four forces acting on the chain element ds are shown. The chain assumes a concave configuration when surface tension (2σ) is larger than the normal component of the gravitational force per unit length (λg cosθ).
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T 2. 2σds. θ. T 1. λgds. 2X. L. Figure 1 The four forces acting on the chain element ds are shown . The chain assumes a concave configuration when surface tension ( 2 σ ) is larger than the normal component of the gravitational force per unit length ( λg cosθ ). T 2. T 1. 2X. L 0. - PowerPoint PPT Presentation
Transcript of 2X
2X
L
θ
T2
2σds
λgds
T1
Figure 1
The four forces acting on the chain element ds are shown. The chain assumes a concave configuration when surface tension (2σ) is larger than the normal component of the gravitational force per unit length (λg cosθ).
θ
λ g ds
2σ ds
2X
L0T1
T2
Figure 2When the normal component of the gravitational force per
unit length (λg cosθ) exactly balances the force per unit length due to surface tension (2σ), the chain assumes a perfect triangular shape.
L
X
θ
λgds
2σds
T2
T1
Figure 3
When the normal component of the gravitational force per unit length (λg cosθ) is greater than the surface tension (2σ), the chain is pulled outward and assumes a concave configuration.
The equilibrium condition for chain element ds:
dT = λg sinθ ds, (1) ( dT = T2 - T1 )
T dθ + 2σ ds = λg cosθ ds (2)
L
X
θ
λgds
2σds
T2
T1
The governing differential equations for the chain:
dT = λg sinθ ds, (1) ( dT = T2 - T1 )
T dθ + 2σ ds = λg cosθ ds (2)
Deviding Eq. (1) by Eq. (2),
dT/T = dθ (sinθ)/(cosθ - α) (3) (α ≡ 2σ/λg)
Integrating Eq. (3),
T = λg C/(cosθ –α) (4) (C is a constant)
And finally the governing differential equation is:
ds/dθ = C/(cosθ –α)2 (5)
Equations For the Triangular Configuration:
dT = λg sinθ ds, (1) ( dT = T2 - T1 )
2σ ds = λg cosθ ds (2) (2σ = λg cosθ)
Integrating Eq. (1) gives tension along the chain,
T = λg sinθ s (3)
θ
λ g ds
2σ ds
2X
L0T1
T2
