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Two point masses m and M are separated by a distance d. At the point on the line joining them where the net gravitational field is zero, the gravitational potential V equals:

1. V = -(G/d)*(m + M)
2. V = -(G/d)*m
3. V = -(G/d)*M
4. V = -(G/d)*(sqrt(m) + sqrt(M))²

Correct answer: V = -(G/d)*(sqrt(m) + sqrt(M))²

Solution

Field is zero where m/x² = M/(d-x)², so x/(d-x) = sqrt(m/M), giving x = d*sqrt(m)/(sqrt(m)+sqrt(M)) and d - x = d*sqrt(M)/(sqrt(m)+sqrt(M)). Potential V = -G*m/x - G*M/(d-x) = -(G/d)*(sqrt(m)+sqrt(M))*sqrt(m) - (G/d)*(sqrt(m)+sqrt(M))*sqrt(M) = -(G/d)*(sqrt(m)+sqrt(M))².

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