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13.3 Gravitational field of a point mass

Syllabus
9702–2028–2029
Topic
13.3
Level
A2

Learning objectives

13.3.1Derive, from Newton’s law of gravitation and the definition of gravitational• derive, from Newton’s law of gravitation and the definition of gravitational field, the equation g = GM / r 2 for the gravitational field strength due to a point mass13.3.2G = GM / r 2• recall and use g = GM / r 213.3.3G is approximately constant for small changes in height near the Earth’s• understand why g is approximately constant for small changes in height near the Earth’s surface

Derive the point-mass field from force per unit test mass

Place a small test mass m at distance r from a point source mass M. Let F be the gravitational force magnitude and G the universal gravitational constant.

Newton′slaw:F=GMm/r2Newton's law: F = GMm/r²

fielddefinition:g=F/mfield definition: g = F/m

g=(GMm/r2)/m=GM/r2g = (GMm/r²)/m = GM/r²

The test mass cancels, so g depends only on source mass M and location r. Its vector direction is toward M. The same result applies outside a uniform sphere when r is measured from its centre.

G is universal, but g is not: doubling centre distance quarters g. A heavier test object experiences a larger force F=mg, not a larger field strength.

Use inverse-square field strength with centre distance and direction

g=GM/r2,directedtowardM;unitNkg−1=ms−2g = GM/r², directed toward M; unit N kg⁻¹ = m s⁻²

Use source mass M and centre distance r in SI units. Calculate the positive magnitude, then state the radial direction or apply a coordinate sign separately. For altitude h above radius R, use r=R+h.

At r = 1.47 × 10¹¹ m from the Sun (M = 1.99 × 10³⁰ kg), g = (6.67 × 10⁻¹¹)(1.99 × 10³⁰)/(1.47 × 10¹¹)² = 6.14 × 10⁻³ N kg⁻¹, directed toward the Sun.

A point Q at r/2 has four times the field magnitude of a point P at r. If Q and P lie on opposite sides of M, their field vectors point in opposite directions because each points toward M.

Do not use altitude alone or scale g as 1/r. Magnitude comparison and vector-direction comparison are separate marks and both may be required.

Small height changes barely alter Earth's centre distance or field pattern

Near Earth's surface, a vertical height change h is tiny compared with Earth's radius R: h ≪ R. Therefore centre distance changes only from R to R+h by a very small fraction.

gh/g0=[GM/(R+h)2]/[GM/R2]=[R/(R+h)]2≈1whenh≪Rg_h/g_0 = [GM/(R+h)²]/[GM/R²] = [R/(R+h)]² ≈ 1 when h ≪ R

Globally Earth's field lines are radial. Over a small region and small height range on a sphere with very large R, those radial lines are approximately parallel and their spacing changes negligibly, representing approximately constant field strength and free-fall acceleration.

The constant-g model supports local projectile/free-fall motion and ΔE_p≈mgΔh. For satellites or heights no longer negligible relative to R, use g=GM/(R+h)² instead.

g is approximately—not exactly—constant near the surface. The approximation is local; radial lines are not globally parallel and g decreases with sufficiently large altitude.