SME
University Physics I: Mechanics
Gravitation
Use this for “Find g at the International Space Station’s height, 400 km above Earth.”
Chart
g(6,771) ≈ 8.694
Level g₀ = 9.82 meets the curve at R = 6,371
This is an example. Type your own values to start a new problem.
g (m/s²)
=
G
M (kg)
(
r (
)
)
2
R
Planet radius
example
h
Height above the surface
example
g₀
Gravity at the surface
calculated · m/s²
g ÷ g₀
Share of surface gravity
worked out (not typed)
Formulas
r
=
R
+
h
6.771 ×
10
6
=
6.371 ×
10
6
+
(4
×
10
5
)
g₀
=
6.674 ×
10
-11
×
M
R
2
9.8195
=
6.674 ×
10
-11
×
(5.972
×
10
24
)
÷
(6.371 ×
10
6
)
2
g
=
6.674 ×
10
-11
×
M
r
2
8.6936
=
6.674 ×
10
-11
×
(5.972
×
10
24
)
÷
(6.771 ×
10
6
)
2
g
g₀
=
(
R
r
)
2
0.8853
=
(
6.371 ×
10
6
6.771 ×
10
6
)²
Formulas are worked in m. The step-by-step shows the conversions.
Assumptions
•
The planet is a sphere, so its mass pulls as if it all sat at the center; r is measured from there.
•
G = 6.674 × 10⁻¹¹ N·m²/kg²; Earth: M = 5.972 × 10²⁴ kg and R = 6371 km.
•
Outside the planet g falls as 1 ÷ r²: doubling the distance from the center quarters g.
Step-by-step
WE KNOW
h = 400 km, M = 5.972 × 10²⁴ kg, R = 6,371 km
FIND
distance from the center (r), gravity at the surface (g₀), gravity at the height (g), share of surface gravity (g ÷ g₀)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to m).
h = 400 km = 400,000 m (1 km = 1000 m)
R = 6,371 km = 6,371,000 m (1 km = 1000 m)
Step 2 · Find distance from the center (r)
Use r = R + h
Gravity is measured from the center, so add the planet’s radius to the height.
r
=
6.371 ×
10
6
+
(4
×
10
5
)
r
=
6,771,000
m
Step 3 · Find gravity at the surface (g₀)
Use g₀ = 6.674 × 10⁻¹¹ × M ÷ R²
Newton’s
law
of
gravitation
on
one
kilogram
at
the
surface:
F
m
=
GM
R
2
.
g₀
=
6.674 ×
10
-11
×
(5.972
×
10
24
)
÷
(6.371 ×
10
6
)
2
g₀
=
(6.674
×
10
-11
)
×
5.972 ×
10
24
4.059 ×
10
13
g₀
=
3.986 ×
10
14
4.059 ×
10
13
g₀
=
9.8195
m/
s
2
Step 4 · Find gravity at the height (g)
Use g = 6.674 × 10⁻¹¹ × M ÷ r²
The
same
law
with
the
distance
from
the
center,
r
=
R
+
h
,
in
place
of
R
.
g
=
6.674 ×
10
-11
×
(5.972
×
10
24
)
÷
(6.771 ×
10
6
)
2
g
=
(6.674
×
10
-11
)
×
5.972 ×
10
24
4.585 ×
10
13
g
=
3.986 ×
10
14
4.585 ×
10
13
g
=
8.6936
m/
s
2
Step 5 · Find share of surface gravity (g ÷ g₀)
Use g ÷ g₀ = (R ÷ r)²
Divide
the
two
laws:
GM
cancels,
leaving
the
square
of
R
over
r
.
g
g₀
=
(
6.371 ×
10
6
6.771 ×
10
6
)²
g
g₀
=
0.9409
2
g
g₀
=
0.8853
Step 6 · Convert the answers
Change each answer back to the unit you chose.
r = 6,771,000 m = 6,771 km (1 km = 1000 m)
Check
Put the numbers back in. Both sides should match.
6.771 ×
10
6
=
6.371 ×
10
6
+
(4
×
10
5
)
✓
9.8195
=
6.674 ×
10
-11
×
(5.972
×
10
24
)
÷
(6.371 ×
10
6
)
2
✓
8.6936
=
6.674 ×
10
-11
×
(5.972
×
10
24
)
÷
(6.771 ×
10
6
)
2
✓
0.8853
=
(
6.371 ×
10
6
6.771 ×
10
6
)²
✓
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