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Two Spheres a and B of Weight 1000n and 750n Respectively Are Kept as Shown in the Figure..Determine Reaction at All Contact Points 1,2,3 and 4. Radius of a is 400 Mm and Radius of B is 300 Mm - Engineering Mechanics

Answer in Brief

Two spheres A and B of weight 1000N and 750N respectively are kept as shown in the figure..Determine reaction at all contact points 1,2,3 and 4. Radius of A is 400 mm and radius of B is 300 mm

Given  : Two spheres are in equilibrium
W1=1000 N
W2=750 N
rA=400 mm
rB=300 mm 
To find : Reaction forces at contact points 1,2,3 and 4

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BC = BP = 300mm = 0.3m
AP = 400 mm = 0.4 m
AB = AP + BP
    = 0.7m
CO = BC + BO
0.7 = 0.3 + BO
BO = 0.4m

`cos α = (BO)/(AB) = (0.4)/(0.7)`
α = cos-1(0.5714) 
α = 55.1501

Forces R3,R4 and 1000N are under equilibrium at point A 

Applying Lami’s theorem
`(R3)/(sin120) = 1000/(sin (150−alpha)) = (R4)/(sin (90+alpha))`

`(R3)/(sin120) = 1000/(sin (150−55.1501)) = (R4)/(sin(90+55.1501))`
Solving the equations
R3 = 869.1373 N
R4 = 573.4819 N

Forces R1,R2,R3 and 750N are under equilibrium at B
Applying conditions of equilibrium

-R3sin α-750+R2=0 
R2=869.1373 sin55.1501+750  
R2=1463.2591 N (Acting upwards) 
Applying conditions of equilibrium


R1=496.65 N(Acting towards right) Point Force
1. R1 496.65 N(Towards right)
2. R2 1463.2591 N(Towards up)
3. R3 869.1373 N(55.1501o in first quadrant)
4. R4 573.4819 N(30o in second quadrant)
Concept: Condition of Equilibrium for non-concurrent nonparallel general forces
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