Gravitation

The One Thing

Gravitation is a universal attractive interaction whose inverse-square force governs falling objects, planetary motion and satellite orbits. Gravitational potential and energy determine whether an object remains bound to a celestial body or can escape its gravitational field.

Reactions, Processes and Experiments

Key Terms

• Universal Law of Gravitation: The force between two masses is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.
• Gravitational Constant: The constant G in Newton’s law of gravitation; its value is approximately 6.67 × 10^-11 N m^2 kg^-2.
• Gravitational Field: The region around a mass where another mass experiences gravitational force.
• Acceleration Due to Gravity: The acceleration produced in a freely falling object by a celestial body; near Earth’s surface, it is approximately 9.8 m s^-2.
• Gravitational Potential: The work done per unit mass in bringing a small test mass from infinity to a point in a gravitational field without changing its kinetic energy.
• Gravitational Potential Energy: The energy possessed by a mass because of its position in a gravitational field; for a mass m at distance r from a mass M, U = -GMm/r.
• Orbital Velocity: The tangential speed required for a satellite to remain in a circular orbit around a celestial body; near Earth’s surface, v_o = √(GM/r).
• Satellite: An object that revolves around a planet or another 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 about 24 hours and moves in Earth’s rotational direction.
• Escape Speed: The minimum initial speed required for an object to escape the gravitational field of a celestial body; v_e = √(2GM/R).
• Kepler’s Laws: Three laws describing planetary motion: orbits are elliptical, equal areas are swept in equal times, and the square of the period is proportional to the cube of the orbit’s semi-major axis.
• Weightlessness: The condition in which the normal reaction on an object becomes zero, such as during free fall or while orbiting in a spacecraft.
• Shell Theorem: A spherically symmetric shell attracts an external object as though all its mass were concentrated at its centre, while the net gravitational field inside the shell is zero.
• Gravitational Potential Reference: Gravitational potential is usually defined by taking its value at infinity as zero; its numerical value therefore depends on the chosen reference level.
• Gravitational Field Strength: The strength of a gravitational field decreases with distance according to an inverse-square relationship.
• Orbital Period: The time taken by a satellite to complete one orbit; for a circular orbit, T = 2π√(r^3/GM).
• Total Mechanical Energy: The sum of kinetic and potential energy; for a satellite in a circular orbit, E = -GMm/(2r).

Easily Confused

• Gravitational potential and gravitational potential energy: Potential is work done per unit mass, measured in J kg^-1 or m^2 s^-2; potential energy is the energy of a particular mass and is measured in joules.
• Orbital velocity and escape speed: Orbital velocity keeps an object in a bound orbit, whereas escape speed allows it to reach infinity with zero final speed.
• Weightlessness and absence of gravity: Weightlessness means that the normal reaction is zero; it can occur during free fall or orbital motion even though gravity is acting.
• Orbital speed in lower and higher orbits: A lower orbit has greater orbital speed and a shorter period than a higher orbit.
• Geostationary and ordinary satellites: A geostationary satellite must have a circular equatorial orbit, eastward motion and a period equal to Earth’s rotational period; these conditions are not required of every satellite.
• Gravitational force and centripetal force: Gravitational force is the physical attraction between masses; in a circular satellite orbit, it supplies the centripetal force.
• Gravitational potential energy and total orbital energy: Gravitational potential energy is U = -GMm/r, whereas total mechanical energy in a circular orbit is E = -GMm/(2r).
• Distance from Earth’s centre and height above Earth’s surface: The orbital formula uses the distance r from Earth’s centre; height above the surface is represented by h, with r = R_E + h.

What Gets Asked

• State or apply Newton’s law of gravitation: Use F = Gm1m2/r^2, with r as the distance between the centres of the masses. Marks are lost by using surface separation instead of centre-to-centre distance or by omitting that the force is attractive.
• Calculate gravitational acceleration at a surface, height or depth: Use g = GM/r^2, g_h = g[R_E/(R_E + h)]^2, or g_d = g(1 - d/R_E) as appropriate. Marks are lost by using the height expression for a depth question.
• Calculate gravitational potential or potential energy: Use V = -GM/r and U = -GMm/r, taking infinity as zero. Marks are lost by omitting the negative sign, which indicates a bound state relative to infinity.
• Derive or calculate orbital velocity and period: Equate gravitational force to centripetal force using GMm/r^2 = mv^2/r, then use v_o = √(GM/r) and T = 2π√(r^3/GM). Marks are lost by confusing orbital radius with height above the surface.
• Explain escape speed: Use v_e = √(2GM/R) = √(2gR) and, for the same body, v_e = √2 v_o. Marks are lost by treating escape speed as dependent on the escaping object’s mass or by confusing it with orbital speed.
• Describe geostationary satellites or apply Kepler’s laws: State the required circular equatorial orbit, eastward direction and approximately 24-hour period for geostationary motion, or identify elliptical orbits, equal areas in equal times and T^2/a^3 constant for Kepler’s laws.