A smooth circular cylinder of weight Q and radius r is supported by two semicircular cylinders each of the same radius r and weight Q/2 as shown in the figure below. If the coefficient of static friction between the flat faces of the semicircular cylinders and the horizontal plane on which they rest is u= and friction between the cylinders themselves is neglected, determine the maximum distance b between the centers B and C for which equilibrium will be possible without the middle cylinder touching the horizontal plane.

 A smooth circular cylinder of weight Q and radius r is supported by two semicircular cylinders each of the same radius r and weight Q/2 as shown in the figure below. If the coefficient of static friction between the flat faces of the semicircular cylinders and the horizontal plane on which they rest is u= and friction between the cylinders themselves is neglected, determine the maximum distance b between the centers B and C for which equilibrium will be possible without the middle cylinder touching the horizontal plane.

SOLUTION:

Since the radius of the three cylinders are same. So, ABC will be a bilateral triangle.

Cos = (b/2)/2r=b/4r

Sin = √1- (b 2 /16r 2 ) = (√16r 2 -b 2 )/4r

From the above figure,

N=Q/2 + R b sinθ

F 1 =R b cosθ

=> N× u = R b cosθ

=>( Q/2 + R b sinθ) ×1/2 = R b cosθ

=> Q/4 + R b × (√16r 2 -b 2 )/8r = R b × b/4r

=> R b ( (2b-√16r 2 -b 2 )/8r)=Q/4 ______(1)

From figure,

R c cosθ = R b cosθ

=> R c = R b

Q= R c sinθ+ R b sinθ

=> Q= 2 R b sinθ( R c = R b )

=> Q = 2× Q/4 × (8r)/2b-(√16r 2 -b 2 ) × (√16r 2 - b 2 )/4r ____ ( from 1)

=> 8b - 4√16r 2 -b 2 = 4√16r 2 -b 2

=> 64b 2 = 64 × ( 16r 2 - b 2 )

=> b 2 = 16 r 2 - b 2

=> 2b 2 = 16r 2

=>  b = √8 × r

=> b = 2.83 r