Gravitational Field Strength Equation (College Board AP® Physics 1: Algebra-Based)

Gravitational field strength equation

• The magnitude of the gravitational field, , created by a system of mass, , at a point in space is equal to the ratio of the gravitational force, , exerted by the system on a test object of mass, , to the mass of the test object,

• Consider the Moon in orbit around the Earth:

  • The Earth is the system of mass that creates the gravitational field of magnitude

  • The Moon is the test object of mass within the Earth's gravitational field

  • Hence, is equal to the ratio of and

Gravitational field strength of Earth on Moon

• This defines the gravitational field strength formula:

• Where:

  • magnitude of the gravitational field or gravitational field strength, measured in

  • gravitational force, measured in

  • mass of test object, measured in

• This is also equal to:

• Where:

  • universal gravitational constant

  • the mass of the system creating the gravitational field, measured in

  • the radius of the system creating the gravitational field, measured in

Derived equation

• The magnitude of the gravitational field, , created by a system of mass, , at a point in space is equal to the ratio of the gravitational force, , exerted by the system on a test object of mass, , to the mass of the test object,

Derivation:

Step 1: Identify the fundamental principles

• Newton's second law

• Newton's law of gravitation

Step 2: Apply the specific conditions

• Newton's second law

  • When the gravitational force is the only force exerted on an object the acceleration, , is equal to the magnitude of the gravitational field,

  • Hence,

• Newton's law of gravitation

  • The mass of the system creating the gravitational field is which represents

  • The mass of the test mass in the gravitational field is which represents

Step 3: Combine and simplify the two conditions

• The gravitational force from Newton's second law equals the gravitational force from Newton's law of gravitation

• Divide by the test mass

  

• Simplify