Introduction:
When a particle of mass m is placed at a distance d from one end of a uniform rod with length L and mass M, there will be a gravitational force acting on the particle due to the rod. In this explanation, we will calculate the magnitude of this gravitational force.

Understanding the Problem:
To find the magnitude of the gravitational force on the particle due to the rod, we need to consider the gravitational attraction between the particle and each infinitesimally small mass element of the rod. By integrating the contributions of all these small mass elements, we can determine the total gravitational force.

Deriving the Formula:
- Consider an infinitesimally small mass element dm of the rod located at a distance x from the end of the rod where the particle is placed.
- The mass of this infinitesimally small element can be calculated using the mass per unit length, which is M/L. Hence, dm = (M/L) * dx.
- The distance between the particle and this infinitesimally small mass element is (d - x).
- The gravitational force between the particle and this small mass element can be calculated using Newton's law of universal gravitation: F = (G * m * dm) / (d - x)^2, where G is the gravitational constant.
- Substituting the value of dm, we get: F = (G * m * (M/L) * dx) / (d - x)^2.

Integration to Find Total Force:
- To find the total gravitational force, we need to integrate the force contributions from all the infinitesimally small mass elements of the rod.
- The total gravitational force can be obtained by integrating the above expression from x = 0 to x = L.
- Let's denote the total gravitational force as F_total.
- Integrating the expression, we get: F_total = ∫ [(G * m * (M/L) * dx) / (d - x)^2] from x = 0 to x = L.

Simplification and Calculation:
- Simplifying the expression and evaluating the integral, we get: F_total = (G * m * M/L) * [1/(d - 0) - 1/(d - L)].
- Further simplifying, we obtain: F_total = (G * m * M/L) * [1/d - 1/(d - L)].
- Rearranging the terms, we get: F_total = (G * m * M) * [(1/L) * (1/d - 1/(d - L))].

Final Result:
Therefore, the magnitude of the gravitational force on the particle due to the rod is given by the expression: F_total = (G * m * M) * [(1/L) * (1/d - 1/(d - L))].

Conclusion:
In this explanation, we derived the formula for calculating the magnitude of the gravitational force on a particle due to a uniform rod. By considering the gravitational attraction between the particle and each infinitesimally small mass element of the rod and integrating these contributions, we obtained the final expression for the total gravitational force.

GmM/d(d+L)