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13.4 Gravitational potential

Syllabus
9702–2028–2029
Topic
13.4
Level
A2

Learning objectives

13.4.1Gravitational potential at a point as the work done per unit mass in• define gravitational potential at a point as the work done per unit mass in bringing a small test mass from infinity to the point13.4.2Φ = –GM / r for the gravitational potential in the field due to a point mass• use ϕ = –GM / r for the gravitational potential in the field due to a point mass13.4.3The concept of gravitational potential leads to the gravitational potential• understand how the concept of gravitational potential leads to the gravitational potential energy of two point masses and use EP = –GMm / r

Gravitational potential is work done per unit mass in bringing a test mass from infinity

Gravitational potential at a point is the work done per unit mass by an external agent bringing a small mass from infinity to that point, with zero potential at infinity.

For an attractive field around a point mass, potential is negative because the field releases energy as the mass moves inward.

Moving a test mass farther from a planet raises its gravitational potential toward zero.

Potential is energy per unit mass, not force; equal potential does not imply equal field strength everywhere.

Use negative inverse-distance potential for a point mass

φ=−GM/r,unitJkg−1;φ(∞)=0φ = −GM/r, unit J kg⁻¹; φ(∞)=0

Use source mass M and centre distance r in SI units. Keep the negative sign: it records the chosen zero at infinity and attractive binding, not a negative magnitude of force.

For Mars, M = 6.4 × 10²³ kg and surface radius r = 3.4 × 10⁶ m. φ = −(6.67 × 10⁻¹¹)(6.4 × 10²³)/(3.4 × 10⁶) = −1.26 × 10⁷ J kg⁻¹.

Outside a spherical source the φ-r curve stays below zero and rises toward zero as r increases. At 2r, φ is half as negative; at 4r, one quarter as negative. Its slope becomes progressively less steep outward.

Potential follows −1/r, not −1/r²; field-strength magnitude follows 1/r². Use r=R+h when altitude is given, and do not let the potential curve cross zero at finite r for an isolated point mass.

Potential energy of two point masses is mass times potential

Gravitational potential φ is potential energy per unit test mass. Therefore a mass m at a point of potential φ has gravitational potential energy E_P=mφ for the two-mass system, with E_P=0 at infinite separation.

EP=mφ=m(−GM/r)=−GMm/rE_P = mφ = m(−GM/r) = −GMm/r

ΔEP=m(φfinal−φinitial);Wexternal(slowmove)=ΔEP;Wgravity=−ΔEPΔE_P = m(φ_final − φ_initial); W_external (slow move) = ΔE_P; W_gravity = −ΔE_P

A 2.63 kg probe at Earth's orbital distance from the Sun, where φ = −9.01 × 10⁸ J kg⁻¹, has E_P = (2.63)(−9.01 × 10⁸) = −2.37 × 10⁹ J.

Slow displacement φ and E_P External work Work by gravity
outward increase toward 0 positive negative
inward become more negative negative positive

E_P is the interaction energy of the mass pair, not an intrinsic property of one isolated object. Negative E_P means positive energy must be supplied to separate a bound pair to infinity; it does not mean energy conservation is violated.