At the point C, there is friction in the direction opposite to F. So friction force must be more than or equal to F, so that the block does not slide forward.
Balancing forces in the horizontal and vertical directions:
N = m g and f >= F
f = force of friction = μ N = μ m g >= F
=> Force F required to topple the block, to make the triangular block rotate :
Let BC make an angle x with the horizontal. So angle ACH = 90 - (60+x) = 30 - x.
The arm length for Force F from pivot C is : CH = a Cos (30-x)
the angle GCJ = 30 +x . So Arm length for weight mg : GC * Cos(30+x)
GC = 2/3 CE = 2/3 * √3/2 * a = a/√3
Hence, balancing the torques/moments of forces:
F * a Cos (30-x) = mg *
* Cos (30+x)
Force F needed to lift the triangular block and make it rotate / topple:
x has range: 0 <= x < 60.
For x > 60, The weight mg and F both are in the same direction. So the block will topple.
= 1/√3 when x = 0
= 1/√3 * Cos50/ Cos10 for x = 20 deg
= 1/√3 * Cos45/Cos15 for x = 15 deg
= 1/√3 * Cos55/Cos5 for x = 25 deg
= 1/2 for x = 30 deg
= 1/√3 * Cos 75/cos 15 for x = 45 deg
= 0 for x = 60 deg.
Force F required to rotate , lift the triangular block is highest when x = 0.
This we want the minimum friction to be more than maximum force F required. Hence, μ >= F / mg => μ >= 1/√3