Question

The 2-dimensional object of art shown in figure is held in display at a museum by support S and rope R. Support S is on the floor and rope R is attached to the ceiling. If the coefficient of static friction between the object and support S is 0.5, what is the maximum value that θ can have? The object's mass is 25 kg and the distances from object's center of mass, CM, to its contact points with the support and the rope are s = 35 cm and r = 55 cm, respectively.

Answer 1

This is a case of static equilibrium.
The net force and the net torque is zero
Take the force of pull from the rope as R
Equating the vertical forces,
fs x cos(30) + R cos $\theta$ = mg
Substituting fs = _$\mu$ mg,
R cos $\theta$ = mg[1 - _$\mu$ cos(30)] ...(1)

fs and s are perpendicular, so the torque from fs is fs x s
R sin $\theta$ = fs x s = _$\mu$ mg x s ...(2)
(2)/(1) gives
tan( $\theta$ ) = [ _$\mu$ x s/r] / [1 - _$\mu$ cos(30)]
By substituting s = 0.35 m, r = 0.55 m,   _$\mu$ = 0.5,
$\theta$ = 29.3^o.