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