A straight rod of length L lies along the x-axis from x = a to x = L + a, with linear mass density lambda(x) = A + B*x². Find the magnitude of the gravitational force it exerts on a point mass m located at x = 0.

G*m*[ A*(1/a + 1/(a+L)) - B*L ]
G*m*[ A*(1/a - 1/(a+L)) + B*L ]
G*m*[ A*(1/a - 1/(a+L)) - B*L ]
G*m*[ A*(1/a - 1/(a+L)) + B*(a+L) ]

Correct answer: G*m*[ A*(1/a - 1/(a+L)) + B*L ]

Solution

Element at x: dm = (A + B*x²) dx, force dF = G*m*(A + B*x²)/x² dx. Integrate from a to a+L: integral of A/x² dx = A*(1/a - 1/(a+L)); integral of B dx = B*L. So F = G*m*[A*(1/a - 1/(a+L)) + B*L].

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