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Circular Motion overview

Topic 4 of 5

Gravity and the near-Earth field

Gravitational attraction becomes weaker with increasing separation. Near Earth's surface, ordinary changes in height are small compared with Earth's radius, so treating the local field as constant is often a good approximation.

Newtonian gravitation uses centre separation

Two point masses m1 and m2, separated by distance r, attract each other with force magnitude:

F = Gm1m2/r2

The force acts along the line joining the masses, towards the other mass. Each body experiences an equal and opposite force; these act on different bodies and do not balance on either individual body.

G is the universal gravitational constant, with units N m2 kg-2. For a point outside a spherically symmetric body, use its whole mass as concentrated at its centre. The separation r is then measured from that centre, not from its surface.

At fixed masses, doubling separation makes the force one quarter as large. At fixed separation, doubling either mass doubles the force on each partner. These statements follow from the inverse-square dependence on r and the product of the two masses.

From gravitational force to field strength

Gravitational field strength is force per unit test mass at a point. Around a spherical source of mass M, for a test mass m outside the source:

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

This expression gives the field's magnitude; its direction is towards the source centre. The test mass cancels, so two different small test masses at the same point experience the same field strength, though their forces mg differ. Unlike the universal constant G, g depends on location.

Using supplied rough scales GM about 4 × 1014 m3/s2 and Earth radius R about 6 × 106 m gives GM/R2 of order 10 N/kg. This is a size check using stated inputs, before the more precise model below.

Worked near-surface comparison

A 100 m height change is small compared with Earth's radius

Model Earth as spherical, neglecting its rotation and the gravity of other bodies. Use the supplied values:

G = 6.67 × 10-11 N m2 kg-2
M = 5.97 × 1024 kg
R = 6.37 × 106 m
GM = 3.98199 × 1014 m3/s2

At the surface, r = R. At height h above it, r = R + h:

gsurface = GM/R2 = 9.81344 N/kg
g100 m = GM/(R + 100)2 = 9.81313 N/kg

The extra digits show a decrease of only about 0.00314%. Both values round to 9.81 N/kg at three significant figures. The field is not exactly constant: the change is simply very small over this height compared with R.

Within a small near-surface region, the inward directions towards Earth's distant centre are also nearly parallel, giving an approximately uniform downward field. Across much larger heights or regions, reassess that approximation.

Outside a spherical source, field magnitude follows an inverse square

R is the source's surface radius; r is distance from its centre. This plot shows only r ≥ R. Its horizontal scale starts at r/R = 1, the surface, and shows no interior field.

Normalized gravitational field magnitude outside a spherically symmetric bodyThe horizontal axis is centre distance divided by source radius, r over R, starting at one and ending at four. The vertical axis is gravitational field magnitude g of r divided by its surface value g of R. For the stated external spherical-source model the curve is one divided by the square of r over R. Highlighted values are one at one radius, four ninths at 1.5 radii, one quarter at two radii, one ninth at three radii and one sixteenth at four radii. The smooth decreasing curve stays positive. No interior or zero-radius curve is drawn, and field direction is not encoded as a negative magnitude.123400.250.50.751g(r) / g(R)Centre distance r / R

At 2R the field is one quarter of its surface value, not one half. At 3R it is one ninth; at 4R it is one sixteenth. The field continues to decrease towards zero as distance grows, but does not reach zero at a finite radius in this model.

The graph shows the magnitude of the external spherical field, divided by its surface value g(R). The horizontal quantity is centre distance divided by radius R. It applies at and outside the surface, r/R at least 1, and does not describe Earth's interior.

At centre distances R, 2R, 3R and 4R, the field magnitudes are g(R), g(R)/4, g(R)/9 and g(R)/16. The inverse-square curve falls steeply at first and becomes flatter as distance increases. It approaches zero with increasing distance; it does not reach zero at a particular finite orbital height.

Field strength and free-fall acceleration

If gravity is the only force on a body, Newton's second law gives:

Fresultant = mg = ma
a = g, directed inward

The field strength in N/kg is then equal in magnitude to free-fall acceleration in m/s2. The units are equivalent because 1 N = 1 kg m/s2, but the definitions differ: one is force per unit mass, the other is change of velocity per unit time.

Support or drag changes the resultant force and acceleration without removing the local gravitational field. A body resting on a table can have zero acceleration while still experiencing weight mg.

An orbiting person and spacecraft can both be in free fall, with no usual support force between them. The person can feel weightless even where gravitational attraction is substantial. Weightlessness in this sense is not evidence for zero g; the orbit calculation uses gravity to explain the curved motion.