5.4 gravity

gravity

a universal attractive force exerted by all objects with mass towards their centres of mass
very weak but infinite range (decreases with distance)

centre of mass

average position of the mass of an object, objects with mass can be modelled as a single point at the centre of mass

gravitational field lines- radial

lines converge into the centre of mass and are labelled with arrows, lines are closer the stronger the field and never cross each other

gravitational field lines- uniform

lines are parallel and spaced at equal intervals, showing the uniformity of the force
shows a small section of the surface of the gravitational object, such as a field on earth

gravitational field strength equation using object in field

force experienced by object in field(N)/mass of object(kg)
accurate assuming the mass of the object in the field is negligible compared to the mass of the object causing the field

Newton’s law of gravitation

two point masses attract each other with a force directly proportional to the product of their masses, and inversely proportional to the square of their separation
F = -GMm/r^2
where F = force(N), G = UGC, M = mass of object 1(kg), m = mass of object two(kg) and r=radius(m)

universal gravitational constant

6.67 × 10^−11 Nm^2kg^-2

gravitational field strength equation using object itself

as g=F/m and F = -GMm/r^2 g can also be given as GM/r^2 where M is the mass of the object itself

Kepler’s first law

the orbit of a planet is an ellipse with its star at one focus (however the ellipse is so close to circular that it can be modelled as such)

Kepler’s second law

a line segment joining a planet and its star will cover the same amount of area per period time as the planet orbits, as the planet moves faster while closer to the star

Kepler’s third law

the square of a planet’s orbital period (T) is proportional to the cube of the average distance (r) from its star

proving Kepler’s third law

centripetal force, provided by gravity, is required to keep the planet in orbit, and this force is provided by the gravitational field of the sun. because of this, we can equate the formula for centripetal force with the formula for gravitational force to give
F = mv^2/r = GMm/r^2
which gives
GM/r = v^2
as the velocity of an object in circular motion is 2pir/t this gives
GM/r = 4pi^r^2/t^2
which gives
t^2 = 4pi^2r^3/GM

satellites

objects that orbit larger objects

geostationary satellites

satellites that stay above the same point on Earth by orbiting:
-in the same direction as Earth’s orbit
-at a speed such that it orbits in one day exactly
-along the equatorial plane

gravitational potential

the work required per unit mass (joules/kg) to move a body to a specific point relative to a massive object from an infinite distance
Vg = -(GM)/r
where G is the gravitational constant, M is the mass of the non-moving object and r is the separation distance between the object and the body

gravitational potential energy

like gravitational potential energy, but total work done, rather than work done per unit mass
E = mVg = -(GMm)/r

escape velocity

for an object to escape a gravitational field, its initial KE must be equal to or greater than the GPE required to lift it to infinity- this is independent of mass
1/2mv^2 = (GMm)/r so v = √((2GM)/r)