Gravitation

The One Thing

Gravitation is the universal attractive interaction between masses, and its strength depends on the masses involved and the distance between their centres. The same gravitational principles explain falling bodies, planetary motion, satellite orbits, gravitational energy, and escape from a gravitational field.

Reactions, Processes and Experiments

Key Terms

• Universal law of gravitation: The force between two point masses is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.
• Gravitational force: The attractive force acting between two masses, given by F = Gm1m2/r^2.
• Gravitational constant: G is the universal constant of gravitation, approximately 6.67 x 10^-11 N m^2 kg^-2.
• Gravitational field: The region around a mass in which another mass experiences gravitational force.
• Gravitational field intensity: The gravitational force experienced per unit mass placed at a point; near a spherical body, g = GM/r^2.
• Acceleration due to gravity: The acceleration produced in a freely falling object by a planet's gravitational field. Near Earth's surface, its average value is about 9.8 m s^-2.
• Mass and weight: Mass is the amount of matter and remains constant, while weight is the gravitational force on an object, W = mg, and can change from place to place.
• Gravitational potential energy: The energy associated with the position of a mass in a gravitational field; taking infinity as zero, U = -GMm/r.
• Gravitational potential: The potential energy per unit mass at a point, V = -GM/r.
• Escape velocity: The minimum speed required for an object to escape a body's gravitational field without further propulsion, ve = sqrt(2GM/R) = sqrt(2gR).
• Orbital velocity: The speed required for a satellite to remain in a circular orbit, vo = sqrt(GM/r).
• Satellite: An object that revolves around a planet or other celestial body under the influence of gravity.
• Geostationary satellite: A satellite that appears stationary above a point on Earth's equator because it has an orbital period of 24 hours, moves west to east, and follows a circular equatorial orbit.
• Kepler's first law: Each planet moves in an elliptical orbit with the Sun at one focus.
• Kepler's second law: The line joining a planet to the Sun sweeps out equal areas in equal intervals of time.
• Kepler's third law: The square of a planet's orbital period is proportional to the cube of the semi-major axis of its orbit: T^2 proportional to a^3.

Easily Confused

• Mass and weight: Mass is the amount of matter and remains constant; weight is W = mg and changes when the local value of g changes.
• Gravitational field and gravitational field intensity: A gravitational field is the region in which a mass experiences gravitational force; field intensity is the force per unit mass at a point.
• Gravitational potential energy and gravitational potential: Gravitational potential energy refers to a mass in a field, U = -GMm/r; gravitational potential is energy per unit mass, V = -GM/r.
• Escape velocity and orbital velocity: Escape velocity is the minimum speed needed to leave the gravitational field without further propulsion; orbital velocity is the speed needed to remain in a circular orbit.
• Geostationary and polar satellites: A geostationary satellite remains apparently fixed above the equator, whereas a polar satellite passes over or near the poles.
• Height and depth variation of g: At height h, g_h = g[R/(R+h)]^2; at depth d, assuming uniform density, g_d = g(1 - d/R).
• Orbital radius and orbital height: Orbital radius is measured from Earth's centre, whereas height is measured above Earth's surface; for a satellite at height h, r = R + h.
• Kepler's laws: The first law concerns the elliptical shape of the orbit, the second concerns areas swept out in equal times, and the third relates orbital period to semi-major axis.

What Gets Asked

• State or apply Newton's law of gravitation, using F = Gm1m2/r^2. Marks are lost by using the distance from a mass's surface instead of the distance between the centres.
• Calculate how g changes with height, depth, or latitude. Marks are lost by treating g as exactly constant everywhere or by using the height formula for a depth question.
• Distinguish mass from weight and apply W = mg. Marks are lost by claiming that mass changes when the local gravitational field changes.
• Derive or apply satellite-orbit relationships, including GMm/r^2 = mv^2/r, vo = sqrt(GM/r), omega = sqrt(GM/r^3), and T = 2pi sqrt(r^3/GM). Marks are lost by confusing orbital radius r with height h.
• Compare orbital and escape velocity. Marks are lost by omitting that escape velocity is sqrt(2) times the circular orbital velocity at the surface or by incorrectly making escape velocity depend on the escaping object's mass.
• Describe satellite types and applications. Marks are lost by omitting the geostationary conditions of a 24-hour period, west-to-east motion, and a circular equatorial orbit, or by assigning polar-satellite applications to geostationary satellites.
• Explain Kepler's laws and gravitational energy. Marks are lost by confusing the three laws or by overlooking the negative sign of gravitational potential energy and total energy for a bound orbit.