Two particles of masses m1 and m2 are connected by a light rod of length r to constitute a dumb-bell.The moment of inertia of the dumb=bell about an axis perpendicular to the rod passing through the center of mass of the two particles is

consider a rod , two particles of mass m₁ , m₂ are at its two ends then

distance of center of mass from m₁ is r₁= m₂r/(m₁+m₂)       ..........1                    (using formula)

distance of center of mass from m₂ is r₂ = m₁r/(m₁+m₂)          .........2

now moment of inertia of two particles is

I₁ = m₁r₁² & I₂ = m₂r₂²

I = I₁+I₂ = m₁r₁² + m₂r₂²            ................3

from 2 & 3 above expression becomes

I = m₁m₂r²/(m₁+m₂)

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also see  question new13 , i have posted the ans

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