SME
University Physics I: Mechanics · Problem type
A circular orbit’s speed and period
Use this for “Find the speed and period of a satellite 400 km above Earth.”
Diagram
Orbital
speed:
v
=
√
GM/r
=
√(6.67
×
10
-11
×
5.97 ×
10
24
/(6.77
×
10
6
))
=
7,670
m/s
Period: T = 2πr/v = 2π × 6.77 × 10⁶/7,670 = 5,550 s (92.4 min)
Gravity is the centripetal force: GMm/r² = mv²/r, so the satellite’s own mass cancels.
It orbits 400,000 m above the surface (r − R, with R = 6.37 × 10⁶ m).
This is an example. Type your own values to start a new problem.
v (
)
=
G
M (kg)
r (
)
R
Planet radius
example
h
Height above the surface
example
T
Period
worked out (not typed)
Formulas
r
=
R
+
h
6.771 ×
10
6
=
6.371 ×
10
6
+
(4
×
10
5
)
v
=
√(6.674
×
10
-11
×
M
r
)
7,672.318
=
√(6.674
×
10
-11
×
5.972 ×
10
24
6.771 ×
10
6
)
T
=
2π
×
r
v
5,545.0579
=
2π
×
6.771 ×
10
6
7,672.318
Formulas are worked in m, s. The step-by-step shows the conversions.
Assumptions
•
The orbit is a circle round the planet’s center, with no air drag; the satellite’s mass is tiny beside M.
•
Only the planet pulls: for Earth this holds out to about 1.5 million km, past which the Sun’s pull wins.
•
G = 6.674 × 10⁻¹¹ N·m²/kg²; Earth: M = 5.972 × 10²⁴ kg and R = 6371 km.
•
Gravity is the whole centripetal force: GMm ÷ r² = mv² ÷ r, so the satellite’s mass m cancels.
Step-by-step
WE KNOW
h = 400 km, M = 5.972 × 10²⁴ kg, R = 6,371 km
FIND
orbit radius (r), orbital speed (v), period (T)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to m, s).
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 orbit radius (r)
Use r = R + h
The orbit circles the planet’s 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 orbital speed (v)
Use v = √(6.674 × 10⁻¹¹ × M ÷ r)
Gravity
supplies
the
centripetal
force:
GM
r
2
=
v
2
r
,
so
v
2
=
GM
r
.
v
=
√(6.674
×
10
-11
×
5.972 ×
10
24
6.771 ×
10
6
)
v
=
√(
3.986 ×
10
14
6.771 ×
10
6
)
v
=
√
5.887 ×
10
7
v
=
7,672.318
m/s
Step 4 · Find period (T)
Use T = 2π × r ÷ v
One
lap
is
the
circumference
2πr
at
speed
v
.
Squared,
this
is
Kepler’s
third
law,
T
2
=
4π²
r
3
GM
.
T
=
2π
×
6.771 ×
10
6
7,672.318
T
=
5,545.0579
s
Step 5 · Convert the answers
Change each answer back to the unit you chose.
r = 6,771,000 m = 6,771 km (1 km = 1000 m)
T = 5,545.0579 s = 92.4176 min (1 min = 60 s)
Check
Put the numbers back in. Both sides should match.
6.771 ×
10
6
=
6.371 ×
10
6
+
(4
×
10
5
)
✓
7,672.318
=
√(6.674
×
10
-11
×
5.972 ×
10
24
6.771 ×
10
6
)
✓
5,545.058
=
2π
×
6.771 ×
10
6
7,672.318
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.