W1*D1 + W2*D2 + W3*D3 + ... + Wn*Dn

This is the TOTAL force on one side.

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(W1*D1 + W2*D2 + W3*D3 + ... + Wn*Dn)/(W1 + W2 + W3 + ... + Wn)

This is the "Weighted Average" distance on one side.

(W1*D1 + W2*D2 + W3*D3 + ... + Wn*Dn)/(D1 + D2 + D3 + ... + Dn)

This is the "Weighted Average" force on one side.

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You can see that they are very closely realated, but somewhat different animals.

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A wieght in the middle is of little consequence. With a distance of 0 from the balance point, the weight is discounted to zero.