PHY C8: Gravitational Field Strength

We’re covering:

• Gravitational Field Strength
• Graph of g against r
• Acceleration of Free Fall

Let’s go!

What is Gravitational Field Strength?
Force per unit mass acting on a small mass placed at a point in the field.

We denote this as a lowercase g.

g = F/m
Since F = GMm/r2,
g = GM/r2

Note that the ‘m’ has disappeared!

• g is INDEPENDENT of the smaller test mass.
• g is the same for 2 small objects at an equal distance from the large mass (M),
even if they have different masses (mA & mB).

Also note that F = ma
g = F/m = a
Thus,
g has a unit of acceleration (ms-2).

When talking about large masses such as planets, we must remember to differentiate between the DISTANCE TO THE CENTRE (r) & the HEIGHT (h).

A bit of notation:

• r = Distance between larger & smaller point mass
• R = radius of planet
• h = height above planet (altitude)

Thus, you can calculate r with:
r = height of m above the surface + radius of planet
r = R + h

For example:
A satellite orbits at 1000 km above the Earth.
The Earth’s radius is 6.4 x 106 m

For calculations, use
r = h + R
R = 1.0 x 106 + 6.4 x 106
R = 7.4 x 106 m

How does g vary with distance from the centre of a large mass?

 Below the surface g increases as distance increases. g ∝ r This is because as you travel further towards the surface, there is MORE MASS beneath you. The larger mass exerts a stronger gravitational force. Below the surface, remember that the point mass is NOT the same as M above the surface. On (near) the surface g is proportional to the mass, & inversely proportional to the square of the radius of the planet. g = GM/R2 For small changes in height near the surface, g is approximately constant (see below). Above the surface g is proportional to the mass, & inversely proportional to the square of the distance to the centre of the planet. g = GM/r2 g = GM/(R + h)2

Field Strength on the Surface of a Planet
g changes noticeably when r changes by a large amount (multiple kilometres),

For small changes in height near a planet’s surface (few metres), the change in g is negligible.
Thus, we can approximate field strength to be constant near the surface.

Since g is in units of acceleration, we also call it the acceleration of free fall.

On Earth, g = 9.81 ms-2.