174-177 - Gravitational Field Strength and Universal Gravitation

A Field Is Where Mass Feels Force

Imagine an astronaut drifting near a planet. The astronaut does not need to touch the planet to be pulled. The useful model is that the planet has set up a gravitational field in the space around it, so any mass placed in that region experiences a gravitational force.

Two masses have different roles in this story:

The source mass creates the field. The test mass reveals it. Removing the test mass does not remove the field around the source mass.

For the equations in this lesson, the force is gravitational attraction. Near a single source mass, the force on a test mass is towards the source mass. The equations usually give the magnitude of the force or field strength; if a direction is needed, state it in words.

Common misconception contrast:

For Edexcel, "state what is meant by a gravitational field" needs the force idea. "The area around a planet" is too vague unless it says that a mass experiences a force there.

[DIAGRAM: asset_name: Inward radial gravitational field around a spherical mass; asset_slug: 051_gravitational_field__diagram_01; recommended_method: matplotlib; description: Inward radial gravitational field around a spherical mass; outside it g relative to surface value follows inverse square distance from centre.]

Field Strength Is Force Per Kilogram

Once a field can exert a force on a mass, the next question is: how strong is the field at this point?

Gravitational field strength is the gravitational force per unit mass placed at a point:

The unit follows directly from the definition: newtons per kilogram. Since $1\,\mathrm{N} = 1\,\mathrm{kg\,m\,s^{-2}}$, $\mathrm{N\,kg^{-1}}$ is equivalent to $\mathrm{m\,s^{-2}}$. In this field lesson, $\mathrm{N\,kg^{-1}}$ keeps the meaning visible: force for each kilogram.

Worked example:

An astronaut of mass $72\,\mathrm{kg}$ has a gravitational force of $115\,\mathrm{N}$ on the Moon. Calculate the gravitational field strength at the astronaut's position.

Decision point 1: the question gives force on a known mass, so use the definition of field strength:

Decision point 2: both quantities are already in SI units, so substitute directly:

Answer:

The interpretation matters: each kilogram of mass at that point would experience about $1.6\,\mathrm{N}$ of gravitational force.

Now double the astronaut's mass while keeping the same place on the Moon. The gravitational force would double, but $F/m$ would stay the same. A bigger mass feels a bigger force; it does not mean the field strength there is bigger.

For an Edexcel “calculate” item, show the equation, substitution and unit. For an “explain” item, link bigger force and bigger mass to the ratio $F/m$, not just to "gravity is the same".

Newton's Law Between Two Masses

The definition $g = F/m$ tells us how strong a field is once we know the force on a mass. Newton's law of universal gravitation tells us the force between two masses:

The distance $r$ is not the gap between surfaces unless the question says the masses are point-like at those positions. For a planet and a spacecraft, use centre-to-centre distance. That usually means planet radius plus height above the surface.

The equation gives the magnitude of the force. The direction is attractive: each mass is pulled towards the other along the line joining their centres. The two forces are equal in size and opposite in direction.

Worked example:

A $720\,\mathrm{kg}$ spacecraft is $400\,\mathrm{km}$ above the surface of Earth. Model Earth as a point mass at its centre. Use: