Final answer:
The derived unit of gravitational potential is joules per kilogram (J/kg). It is calculated using the formula V = -G * M / r, where G is the gravitational constant, M is the mass creating the gravitational field, and r is the distance from the mass to the point of measurement.
Step-by-step explanation:
The gravitational potential (not to be confused with gravitational potential energy), is a measure of the potential energy per unit mass at a point in a gravitational field. The gravitational potential at a point in space is defined as the work done by an external force to bring a mass from infinity to that point, without any acceleration. Since it's a form of potential energy per unit mass, its derived unit is joules per kilogram (J/kg).
The formula to calculate the gravitational potential (V) due to a mass M at a distance r is given by:
V = -G * M / r
where G is the gravitational constant (6.674 × 10^-11 N*m²/kg²) and r is the distance from the center of the mass M to the point where the potential is being measured.
Using the dimensional analysis:
- Gravitational constant (G) has units N · m²/kg² (newton meter squared per kilogram squared).
- M (mass) has units kg (kilograms).
- r (distance) has units m (meters).
So the derived unit for gravitational potential is:
J/kg = (N · m · m²) / (kg² · m)
Simplified, N · m = J (joule), and kg²/kg simplifies to kg, leaving us with J/kg.