2.6 Gravitational Force

Describe the gravitational interaction between two objects or systems with mass.

• Newton’s law of universal gravitation describes the gravitational force between two objects or systems as directly proportional to each of their masses and inversely proportional to the square of the distance between the systems’ centers of mass. Relevant equation: FG r g 12 2= mm
  • i. The gravitational force is attractive.
  • ii. The gravitational force is always exerted along the line connecting the center of mass of the two interacting systems.
  • iii. The gravitational force on a system can be considered to be exerted on the system’s center of mass.
• A field models the effects of a noncontact force exerted on an object at various positions in space. TOPIC 2.6 Gravitational Force
  • i. The magnitude of the gravitational field created by a system of mass M at a point in space is equal to the ratio of the gravitational force exerted by the system on a test object of mass m to the mass of the test object. Derived equation: g F m G M r g 2== 
  • ii. If the gravitational force is the only force exerted on an object, the observed acceleration of the object (in m/s2) is numerically equal to the magnitude of the gravitational field strength (in N/kg) at that location.
• The gravitational force exerted by an astronomical body on a relatively small nearby object is called weight. Derived equation: Fm gWeightg==

Describe situations in which the gravitational force can be considered constant.

• If the gravitational force between two systems’ centers of mass has a negligible change as the relative position of the two systems changes, the gravitational force can be considered constant at all points between the initial and final positions of the systems.
• Near the surface of Earth, the strength of the gravitational field is

Describe the conditions under which the magnitude of a system’s apparent weight is different from the magnitude of the gravitational force exerted on that system.

• The magnitude of the apparent weight of a system is the magnitude of the normal force exerted on the system.
• If the system is accelerating, the apparent weight of the system is not equal to the magnitude of the gravitational force exerted on the system.
• A system appears weightless when there are no forces exerted on the system or when the force of gravity is the only force exerted on the system.
• The equivalence principle states that an observer in a noninertial reference frame is unable to distinguish between an object’s apparent weight and the gravitational force exerted on the object by a gravitational field.

Describe inertial and gravitational mass.

• Objects have inertial mass, or inertia, a property that determines how much an object’s motion resists changes when interacting with another object.
• Gravitational mass is related to the force of attraction between two systems with mass.
• Inertial mass and gravitational mass have been experimentally verified to be equivalent.

Describe the gravitational force exerted on an object by a uniform spherical distribution of mass.

• The net gravitational force exerted on an object by a uniform spherical distribution of mass is the sum of the individual forces from small differential masses that comprise the distribution.
• Newton’s shell theorem describes the net gravitational force exerted on an object by a uniform spherical shell of mass.
  • i. The net gravitational force exerted on an object inside a thin spherical shell is zero.
  • ii. The net gravitational force exerted on an object outside a thin spherical shell can be determined by treating the shell as a single massive object located at the center of the shell.
  • iii. An object inside a sphere of uniform density experiences a net gravitational force from only a partial mass of the sphere.
  • iv. The partial mass of a sphere that contributes to the net gravitational force exerted on an object within that sphere is the portion of the sphere’s mass located a distance less than or equal to the object’s distance from the center of the sphere and can be calculated using the density of the sphere. Derived equation:
• The gravitational force exerted on an object within a uniform sphere can be shown to be proportional to the object’s distance from the sphere’s center. Derived equation: Fk rg,p artial pa rtial=− BOUNDARY STATEMENT AP Physics C: Mechanics does not expect students to mathematically prove or derive Newton’s shell theorem.