Rucete ✏ AP Physics C In a Nutshell
9. Universal Gravitation
This chapter introduces the law of universal gravitation, the principle of superposition, gravitation due to spherically symmetric masses, the relationship between g and G, Kepler’s Third Law, and gravitational potential energy.
Newton’s Law of Universal Gravitation
• Every particle attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.
• The law has the same mathematical form as Coulomb’s law for electric forces.
• In vector form, the gravitational force acts along the line joining two masses.
Gravitation Due to Spherically Symmetric Mass Distributions
• Outside a spherically symmetric mass, gravity acts as if all mass were concentrated at the center.
• Inside such a mass, only the mass enclosed within a given radius contributes to gravitational force.
• Two cases arise: uniform density (simple volume scaling) and variable density (requires integration).
Problem Solving: Gravity Inside Spheres
• For uniform density, enclosed mass is proportional to volume.
• For variable density, enclosed mass requires integration over spherical shells.
• Example problems involve finding gravitational forces for different mass distributions.
Relating g to G
• Near Earth’s surface, gravitational acceleration g = 9.8 m/s² is a valid approximation.
• g is derived from Newton’s law using Earth’s mass and radius.
• At small heights above Earth, g remains approximately constant.
Gravitational Fields
• Gravitational acceleration g is an example of a gravitational field.
• The field connects source masses to the gravitational forces experienced by objects.
• Gravitational fields vary across space and are vector fields assigning a vector to every point.
Kepler’s Third Law
• For planets orbiting the same star, T² is proportional to R³, where T is the orbital period and R is the semimajor axis.
• This relationship arises from equating gravitational force with centripetal force.
• Proof for circular orbits shows that R³/T² is constant for a given central mass.
Gravitational Potential Energy
• Near Earth’s surface, U = mgy is a valid linear approximation.
• More generally, gravitational potential energy is U = -Gm₁m₂/r.
• Potential energy depends only on the initial and final positions, making gravity a conservative force.
Problem Solving: Gravitational Potential Energy
• Outside spherical masses, potential energy behaves as if all mass were concentrated at the center.
• Inside spheres with uniform density, different force laws apply requiring segmented integration.
• Examples include calculating potential energy curves for solid spheres and satellites.
Gravitational Potential Energy for Systems of Masses
• Total potential energy for a mass interacting with several masses is the sum of each individual interaction.
• Superposition principle applies to gravitational potential energy just like it does to forces.
Solving Problems Involving Universal Gravitation
• Useful tools include centripetal force relations, velocity-period-radius equations, gravitational potential energy equations, and conservation laws.
• Angular momentum and total energy conservation are often used in orbital mechanics.
• Correct application of Newton’s and Kepler’s laws allows solving diverse gravitational problems.
In a Nutshell
Universal gravitation governs the attractive forces between masses, influencing planetary orbits and gravitational potential energy. Using principles like superposition and conservation laws, gravitational interactions can be simplified and solved. Kepler’s Third Law provides a powerful relation between a planet’s period and distance from its star, stemming directly from Newton’s gravitational principles.