Knowing that the line of action of the force Q passes through point C, derive an expression for the magnitude of Q required to maintain equilibrium.

Knowing that the line of action of the force Q passes through point C, derive an expression for the magnitude of Q required to maintain equilibrium.

Let us consider the virtual displacement of the linkage as shown in figure:

From the principle of virtual work,

𝛿U = 0

or, -PδYA - QδCD = 0 -------(1)

From figure,

YA = 2lcosΘ

or, δYA = -2lsinΘ

Also, CD = OC + OD

Or, CD = CEsinΘ/2 + DEsinΘ/2

or, CD = 2lsinΘ/2

or, δCD = 2lcosΘ/2δΘ

From eqn 1, we get,
or, -PδYA - QδCD = 0

or, -Px(-2lsinΘ) - Qx(2lcosΘ/2)δΘ = 0

or, 2lPsinΘ = 2lQcosΘ/2

∴ \( Q = \frac{2P \sin \Theta}{\cos \frac{\Theta}{2}} \)