IB recommended length: 5 hours (core) + 7 hours (AHL)
The title of this topic is Gravitational fields but I would argue that there is only one gravitational field, covering all of space-time. That field, like all fields, has values and directions that vary from point to point, as determined by the location of masses in space and time! The motion of objects in freefall is determined by the local values of the gravitational field. In the words of John Wheeler, "Matter tells spacetime how to curve. Spacetime tells matter how to curve." Although Sir Isaac Newton is famous as the 'discoverer' of gravity, he had no concept of gravitational fields or spacetime. In his model of the universe masses acted directly on each other, but it is with Newton we start.
The last great observational astronomer before the adoption of telescopes was Tycho Brahe (1546 - 1601). The Danish royal astronomer was well known for his extremely accurate measurements of the positions of the stars and the planets, as well as for losing his nose in a duel about mathematics and having a pet elk that died after drinking too much beer and falling down the stairs. In 1599 Tycho moved from Denmark to Prague where, in the last year of his life, he started working with an assistant called Johannes Kepler (1571 - 1630). Kepler (unlike Tycho Brahe) believed that the sun, not the Earth, was in the centre of the solar system, and he used the data of Tycho Brahe to investigate the motion of the planets around the sun. From his analysis he derived three empirical (based on observation) laws of orbital motion.
I. The orbit of a planet is an ellipse with the Sun at one of the two foci.
II. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
III. The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit.
In the history of science the first law is important, because it showed that the universe is imperfect ... unlike earlier theories of astronomy that imagined orbits as perfect circles and spheres. (However, note that any questions the IB gives you involving orbits will assume circular orbits.) The second and third laws became important when, over eighty years later, Sir Isaac Newton was able to derive relationships like Kepler's based on his own laws of motion and universal gravitation.
Newton's second law of motion describes how a force acting on an object will change its momentum. Newton's universal law of gravitation is a way of calculating a force on an object, resulting from the gravitational attraction of another object. According to this law a force pair (as described by Newton's third law of motion) exists between any two objects with mass. The direction of the force is always attractive (the objects attract each other) along a straight line connecting the two objects, and the magnitude of the force is proportional to the product of their masses and inversely proportional to the square of the distance between their centres. The constant of proportionality for the equation is known as "big G", the Gravitational constant and has a value of 6.67 x 10^-11 Nm²/kg².
First published in Newton's Principia in 1686, the law is "universal" because, in a radical move for the time, it describes what is happening to objects on Earth (such as apples falling) as well as in the heavens (such as the Moon's orbit around the Earth). Although all objects in the real universe are extended (not point masses), Newton's law assumes point masses or that all objects can be treated as point masses, which is valid either when the distances between objects are much greater than the sizes of the objects, or when dealing with homogenous (or radially symmetric) spheres.
VIRTUAL LAB: Inquiry into Gravitational Force: PhET link and instructions on YouTube
An object placed at any point in a gravitational field will experience a force. The magnitude of the force will be determined by the mass of the object (in kg) and the local gravitational field strength (in N/kg). The standard symbol for gravitational field strength is "little g", g. The average value at the surface of the Earth is g = 9.8 N/kg. This varies from more at the eqautor to less at the poles, as the Earth is a slightly squashed sphere. As an equation, g = F/m and, more generally, g = GM/r², where M is the 'large' mass mostly responsible for the local gravity, r is the distance to the centre of that mass and G is the univeral gravitational constant.
All fields can be represented graphically by field lines. Since our universe has three dimensions of space (and one of time) then the best diagrams are three dimensional (3D). However most diagrams are drawn in two dimensions (2D) as a cross-section through an area of space;
In gravitational field diagrams, the lines represent the direction of force on a "small" test mass located at that point in a field, with the arrow on the line showing what direction along the line is used. The magnitude of the force can be determined using gravitational field strength or Newton's universal law of gravitation. On large, astronomical, scales, gravitational field diagrams are mostly radial, with lines terminating on planets, stars and similarly massive objects. On smaller scales, such as rooms, field lines are vertically orientated as objects fall towards the ground / floor.
Potential energy is energy, measured in joules, due to the position of an object, unlike kinetic energy which is due to the velocity of an object. EP is the symbol for potential energy. For a single object in a gravitational field changes to the potential energy of the object are the result of movement of the object in the gravitational field. Doing work on a rock, against gravity, by picking it up will give gravitational potential energy to the rock. Dropping it will decrease its gravitational potential energy.
A system of objects will have a total gravitational potential energy. That will be found by calculating the work done to assemble the system from infinite initial separation of the objects. Since gravity is always attractive you need to do work to stop the system from assembling, any two objects will naturally come together. This means that the gravitational potential energy of a system is always negative. For a system of two bodies, the equation is: EP = −G(m1m2)/r where r is the separation between the centre of mass of the two bodies.
Just as gravitational field strength is the property of a point in a field that, when multiplied by the mass of an object located at that point will give the force on that object, at that point; so gravitational potential is the energy per unit mass that an object of a given mass will have at that point. The symbol for gravitational potential is Vg and the value of gravitational potential at a point is defined as the work done per unit mass in bringing a mass from infinity to that point as given by Vg = –GM/r. The units of gravitational potential are joules / kilogram (J/kg).
A graph of how the gravitational potential varies between masses can be analysed to find the gravitational field strength, g, at a point along that path. The field strength is equal to the gravitational potential gradient, as given by g = − ∆Vg / ∆r.
Returning to moving a single object in a gravitational field, the work done (W, in joules) moving an object (of mass m, in kilograms) in a gravitational fields is given by W = m ∆Vg . Since the value of the gravitational potential varies with r, the distance to the centre of the significant mass, then only motion towards or away from that centre will cause a change in the work done. Motion perpendicular to this direction, i.e. parallel to the ground or floor, will not do require work, and all of these points have the same gravitational potential. Thus they are known as equipotential surfaces.
Equipotential surfaces are always perpendicular to field lines. In small scale situations, such as in a room or close to the surface of a planet, we assume that the field lines are vertical, and thus the equipotential surfaces / lines are horizontal - parallel to the ground / floor. On larger scales, such as around planets or stars, the equipotential surfaces are in the form of spheres surrounding the body.
A projectile, as described in section A, is an object moving under the influence of gravity (ignoring air resistance). If a projectile is launched upwards from a large mass, such as a planet, then it will slow down as if climbs due to the attraction of the mass. The ultimate fate of the object will be either to slow down to a stop and then fall back to the planet, or keep going forever. The latter will occur if the initial kinetic energy of the object is greater than the magnitude of potential energy of the system. The velocity at this will occur is a function of the mass of the large object and the radius of the object (if launched from the surface): vesc = √(2GM/r)
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Banner By Event Horizon Telescope, uploader cropped and converted TIF to JPG - https://www.eso.org/public/images/eso1907a/ (image link) The highest-quality image (7416x4320 pixels, TIF, 16-bit, 180 Mb), ESO Article, ESO TIF, CC BY 4.0, https://commons.wikimedia.org/w/index.php?curid=77925953