D1 gravitational fields

gravitational field

region of space surrounding a body, in which another body placed in it experiences a gravitational force (attractive) due to their masses

gravitational field near earth’s surface

• field lines parallel and equally spaced → uniform field

• gravitational field strength is constant at 9.8

gravitational field over large distances from earth’s surface

• field lines directed radially towards earth’s centre, increasing spacing between field lines at further points

• gravitational field is non-uniform, weakens with distance from earth’s centre

newton’s law of gravitation

two point masses attract each other with a force that is

• directly proportional to product of their masses

• inversely proportional to the square of their separation

F = Gm₁m₂ / r² (unit: N)

note:

• only for point masses → ie can only use for spheres when separation r is large compared to the radii of the spheres

• there is an equal and opposite ATTRACTIVE force acting on each mass, even if masses are not equal (since it is m₁m₂)

• gravitational force F of earth on satellite provides the centripetal force for satellite to orbit around earth

• F follows direction of field lines

gravitational field strength

(at a point in the gravitational field) the gravitational force exerted per unit mass on a small mass placed at that point

note: it is a vector that acts at a point, even if there is no mass at that point. assumes the source mass (the mass generating the gravitational field) is a point mass.

g = F/m = GM/r² (unit: N kg^-1)

why does a=g?

g = F/m = ma/m = a

note: g is in N kg^-1 but a is in m s^-2, same magnitude but different units so they are different physical quantities

graph of gravitational field strength against distance for a single isolated source mass

note: inverse works too. direction depends on which direction is taken as pos/neg.

graph of gravitational field strength against distance for 2 point masses

note: if masses are different, neutral point will be nearer to the smaller mass

graph of gravitational field strength against distance from centre of mass

• within earth, directly proportional to r

  • derived from mass = density X volume and volume = volume of sphere

• outside of earth, based on formula, inversely proportional to r²

rotational motion

• tangential speed is large enough that it orbits

• to keep moon in orbit, gravitational force of earth on moon acts as centripetal force (net force is towards centre of earth, but velocity direction always changing)

• v_orbit = sqrt (GM/R) where M is mass of earth (HL formula)

  • derived from GMm/r² = mv²/r where m is mass of moon

centripetal force

• F_centripedal = ma = mv²/r = mw²r

  • acceleration taken from A2 data booklet formula

  • where m is mass of moon

  • w (omega) = 2pi/T (where T is the period of rotation)

• T² is directly proportional to R³ (kepler’s 3rd law)

  • derived from F_g = F_centripedal

kepler’s 3rd law

T² is directly proportional to R³

as object is further from source mass, period of oscillation increases

kepler’s 1st law

planetary orbits are elliptical with a star (the sun) at a focus (not centre! focus! remember your math ia)

T² is directly proportional to R³ gives an elliptical graph

in calculations we assume circular path of orbit (but need know this theory)

kepler’s 2nd law

the radius vector from the star (the sun) to the orbiting body sweeps equal areas in equal times

• since nearer, F_g increases, a increases, velocity increases → in the same time, the same area is covered

• if time taken to travel distance 1 = time taken to travel distance 2 → area 1 = area 2