Gravitational potential energy is the energy possessed by an object due to its position in a gravitational field. To calculate the gravitational potential energy of a solid hemispherical object, we need to consider the gravitational potential energy of each infinitesimally small element of the object and integrate it over the entire object.

Gravitational Potential Energy of an Infinitesimally Small Element:
Consider an infinitesimally small element of the hemispherical object with mass dm and radius r. The gravitational potential energy of this element is given by:

dU = -G * (M * dm) / r

Here, G is the gravitational constant.

Integration to Find Total Gravitational Potential Energy:
To find the total gravitational potential energy of the hemispherical object, we need to integrate the above expression over the entire object. Since the mass distribution is symmetric, we can integrate over the limits from 0 to R for the radius and from 0 to π/2 for the angle.

U = ∫∫ (-G * (M * dm) / r) dA

Where dA is the infinitesimal area element on the surface of the hemisphere.

Calculating the Infinitesimal Area Element:
In spherical coordinates, the infinitesimal area element dA is given by:

dA = r^2 * sinθ * dθ * dφ

Here, θ is the polar angle and φ is the azimuthal angle.

Substituting the expression for dA into the integral, we get:

U = ∫∫ (-G * (M * dm) / r) * r^2 * sinθ * dθ * dφ

Simplifying the expression, we have:

U = -G * M * ∫∫ dm * sinθ * dθ * dφ / r

Since the mass distribution is uniform, dm = (M / V) * dV, where V is the volume of the hemisphere.

U = -G * M * ∫∫ (M / V) * dV * sinθ * dθ * dφ / r

U = -G * M^2 * ∫∫ (1 / V) * dV * sinθ * dθ * dφ / r

Finding the Total Volume:
The volume of the hemisphere can be calculated as:

V = (2/3) * π * R^3

Substituting this into the expression for U, we get:

U = -G * M^2 * ∫∫ (1 / [(2/3) * π * R^3]) * dV * sinθ * dθ * dφ / r

Simplifying further, we obtain:

U = (-3GM^2) / (2πR^3) * ∫∫ dV * sinθ * dθ * dφ / r

U = (-3GM^2) / (2πR^3) * ∫∫ r^2 * sinθ * dθ * dφ / r

U = (-3GM^2) / (2πR^3) * ∫∫ r * sinθ * dθ * dφ

Evaluating the Integral:
Integrating over the limits, we find:

U = (-3GM^2) / (2πR^3) * [(2/3) * R^3