Q.Define gravitational potential at a point in a gravitational field. Derive the expression $V(r) = -GM/r$ for the potential at a distance $r$ from a point mass $M$, and explain why the gravitational potential due to any real mass is always negative (with the usual reference at infinity).

Gravitational potential is the work done per unit mass in bringing a test mass from infinity to a point in a gravitational field, given for a point mass M by V = -GM/r.

Every mass creates a gravitational field around it, and every point in that field can be assigned a gravitational potential energy per unit mass, called the gravitational potential.

Definition: The gravitational potential V at a point in a gravitational field is the amount of work done in bringing a unit mass (1 kg) from infinity to that point, without giving it any kinetic energy (i.e., moved slowly/quasi-statically against the field).

Mathematically, if W is the work done in bringing a mass m from infinity to a point, then

V = W/m

For a point mass M (or a spherically symmetric mass, treated as concentrated at its centre), the gravitational potential at a distance r from it is

V = -GM/r

…

Gravitational potential is defined the same way as electric potential, but for gravity: it is the work needed per unit mass to bring a test mass from far away (infinity) to the given point, and works out to V = -GM/r for a point mass (or spherical mass) M.

Definition: The gravitational potential (V) at a point in a gravitational field is the amount of work done in bringing a unit mass (1 kg) from infinity to that point, without giving it any acceleration (i.e., quasi-statically, against the gravitational pull).

For a point mass (or uniform sphere) of mass M, at a distance r from its centre:

V = -GM/r

where G is the universal gravitational constant.

Key points:

• V is a SCALAR quantity (unlike gravitational field, which is a vector).
• SI unit: J/kg (or equivalently, m²/s²). …