SL - Gravitational fields

SL - Gravitational fields

1. Kepler's 3 laws of orbital motion

   • 1st law - elliptical orbits
     • A circular orbit has a single focus at its centre, while an elliptical orbit has two foci, with the star at one of these foci.
       • No planet has a perfectly circular orbit.
     • Each planet orbits the Sun along an elliptical path. The Sun is located at one of the foci of the elliptical orbit.
     • For elliptical orbits, the distance between the two objects changes constantly, while for circular obits this distance is constant.
       • The sum of the distances between an object following an elliptical orbit and the individual foci is constant.
   • 2nd law - equal areas
     • As the planet follows its orbit, the imaginary line joining the Sun and a planet sweeps equal areas of space in equal time intervals.
       • further from star -> longer line -> slower speed
       • closer to star -> shorter line -> faster speed
       • A = CONSTANT
     • elliptical orbit = speed of planet changes, circular orbit = speed of planet is constant
   • 3rd law - T^2 ∝ r^3
     • T^2 ∝ r^3
     • The square of the orbital period is directly proportional to the cube of the average distance from the planet to the Sun.
       • T = orbital period, time it take a body of mass to completely orbit another in seconds, r = average distance from the planet to the Sun in metres
         • Derivation of orbital period, T
           • Orbital period = T, which is the time it takes a body of mass m to complete one orbit of another body of mass M, at a constant speed, v. The distance between the centres of the two bodies is r. 1. Equate the gravitational force of attraction between the two masses to the centripetal (resultant) force acting on the orbitting mass. GMm/(r^2)=m(v^2)/r So, GM/r = mv^2 2. Speed is given by distance covered / time taken, in this case circumference of the circular trajectory taken / time taken for a complete revolution or orbital period. So: v=2πr/T Substituting:GM/r=4(π^2)(r^2)/(T^2) Rearranging: T^2 = 4(π^2)(r^3)/GM This is Kepler's third law.

2. Newton's universal law of gravitation

   • F = Force in Newtons G = gravitational constant in N m^2 kg^-2 m1 and m2 = masses of the two bodies in kg r = distance between the two bodies' centres in metres, m
   • gravity ∝ m1m2 , gravity ∝ 1/(r^2)
   • Every object in the Universe attracts every object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres. The direction of the force is along the line joining them.
     • Example of an inverse square law y = 1/x^2
   • Sometimes written where M = m1 and m = m2
   • This law applies directly to two point masses, or bodies that can be approximated as such.
     • For this approximation to be made, the bodies need to have uniform density - spherical symmetry (here it would treat it as if all the mass were concentrated in the centre of the body), or large separation. Basically, its volume should be negligible.
   • Make sure that you are working with the distance between centres NOT between surfaces of bodies, here you would add radius to get the real r value.
   • Any orbitting object is kept in orbit by the graviational force of attraction, providing a centripetal force that keeps the orbitting oject in its circular trajectory.

3. Gravitational fields and gravitational field lines

   • A gravitational field is any region of space where a mass experiences a gravitational force.
     • They exist around all objects with mass, but are very small for small masses (e.g. humans).
     • Gravitational fields act as a distance, i.e. no contact needed.
     • Gravitational fields have infinite range; every gravitational field extends for infinite distance. However, it becomes weaker with distance from the mass that is the source of the field.
   • We use gravitational field lines to represent gravitational fields. These show the direction of the gravitational force acting on a mass at that point.
     • You can fields that are radial, uniform, etc. a region of space where the field has the same magnitude and direction at every point. This is often visualized by parallel and evenly spaced field lines
     • Gravitational fields are always attractive, hence gravitational field lines point towards the centre of an object. They never cross one another.
     • The closer together the field lines, or the higher the density of field lines, the stronger the gravitational field (stronger force experienced by mass).

4. Gravitational field strength

   • Gravitational field strength, g, at a point in a gravitational field is defined as the gravitational force per unit mass acting on a point mass, m.
   • Like gravitational field density
   • F=ma, but as gravitational acceleration is a type of acceleration, Gravitational force F, is found through F=mg, rearranging to find gravitational field strength, g, gives: g=F/m.
     • If you combing g=F/m with the equation for Newton's universal law of gravitation you obtain: g = GMm/mr^2 = GM/r^2.
   • Gravitational field strength and gravitational acceleration are effectively the same concept, representing the force of gravity per unit of mass, and are therefore numerically equivalent.
   • N/kg
   • 9.8 N/kg at the Earth's surface -> beware this value changes with distance from the Earth's surface
   • Vector
   • If two objects are exerting a gravitational force on a mass hat are equal in magnitude but opposite in direction, there is no resultant gravitational field at that point, gravitational field strength is 0.

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