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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

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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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Solution


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

In △AOB
`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

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

ΣFX=0
R1-R3cosα=0 
R1=869.1373cos55.1501  

R1=496.65 N(Acting towards right)

Sr.no. 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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