Determine the gravitational field produced by a uniform rod of length L and mass M at a point on its perpendicular bisector, positioned at a distance d from the centre.

Step-by-step solution

Consider a small element of length dx at a distance x from the centre of the rod. Mass of the element, dm = M L dx The gravitational field due to this element at the point on the perpendicular bisector is dE = G dm r^2 = G M dx L(x^2 + d^2) Due to symmetry, the components of the gravitational field parallel to the rod cancel out. The net field is along the perpendicular bisector. dE_ net = dE = G M dx L(x^2 + d^2) d x^2 + d^2 = G M d L dx (x^2 + d^2)^ 3/2 Integrating from x = - L 2 to x = L 2 : E = _ -L/2 ^ L/2 G M