HL - Gravitational fields

HL - Gravitational fields

1. Gravitational potential energy, GPE

   • Gravitational potential energy is the energy stored in a mass due to its position in a gravitational field. This energy is what allows the object to do work if it moves to the object exerting an attractive force = where the energy comes from.
     • If we raise an object above ground level, we are moving it away from the direction of the gravitational force pulling it towards Earth. Hence, it gains gravitational potential energy.
       • If we drop the object, as it falls, this gravitational potential energy transfers to kinetic energy (and thermal energy).
       • At the surface of the Earth, gravitational potential energy is NOT ZERO. If an object falls down a hole its GPE decreases further. Gravitational potential energy is zero when an object is not acted upon by gravitational forces at all, i.e. when it is infinitely far away from other masses.
         • If an object is in a gravitational field, its GPE is negative.
     • Deriving a formula:
       • Work done: W=Fs
         • For two small bodies of mass m1 and m2, a distance r apart, if they are moved closer together, and the change in displacement is small, then: ΔEp =~ -FΔs
           • Combining this with the equation for gravitational force, F = GMm/r^2 between two spherical or point masses: ΔEp =~ -GMmΔs/r^2
             • NOT exact but the exact one is outside of IB Physics scope
           • small Δs as assumes uniform gravitational field
       • Only valid for point masses or spheres
       • Close to the Earth's surface we use: ΔEp =mgΔh
         • F=mg close to the Earth's surface, and Δh is Δs.
   • never positive
     • Moving mass m further away from mass exerting a force of attraction requires doing work against the gravitational force acting on m.
       • W=ΔEp
         • Between two points (Ep2 further than Ep1): W=Ep2-Ep1
           • Ep=mVg
             • W=m(Vg2-Vg1)
   • For a pair of masses, the GPE is the work that would be done in bringing them from infinite separation to their current positions.
     • Since GPE decreases, no positive work needs to be done, so work done is negative, and potential energy is negative.
     • Applies to more than 2 objects.
   • If GPE increases, positive work needs to be done.
   • J, joules
   • Scalar

2. Gravitational potential, Vg

   • The work done per unit mass in bringing that mass from infinity to that point in a gravitational field.
   • GPE per unit mass
   • Vg = Ep/m = -GMm/rm = -GM/r
     • in J/KG
   • Scalar
   • Can only take values less than or equal to zero.
   • Proportional to - 1/r, so as r increases, the magnitude of Vg decreases, but Vg technically increases as it becomes less negative.
     • Gravitational potential (Vg) tends to negative infinity as you approach the center of a massive object.
   • We use gravitational potential instead of GPE because potential is an intrinsic property of the gravitational field itself, defined per unit mass, while GPE is specific to a particular object within that field.
   • Look at pink by GPE for explanation: W = mΔVg

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