SME
Flight Mechanics, Stability & Control · Problem type
Load factor and radius of a level turn
Use this for “An airplane flies a level turn at 80 m/s, banked 60°. Find the load factor, the turn radius and the turn rate.”
Diagram
Up
and
down:
L
cos
φ
=
W,
so
n
=
L
W
=
1
cos 60
°
=
2
Level:
L
sin
φ
=
mV²
R
,
the
centripetal
force
R
=
V
2
g tan φ
=
80
2
9.81 × tan 60°
R = 376.7 m
ω
=
V
R
=
0.2124
rad/s
=
12.2°/s
This is an example. Type your own values to start a new problem.
n
=
1
cos(
°)
R (m)
=
(
V (m/s)
)
2
9.81
×
tan(
°)
ω
Turn rate
calculated
Formulas
n
=
1
cos(
φ
)
2
=
1
cos(60°)
R
=
V
2
9.81 × tan(
φ
)
376.7
=
80
2
9.81 × tan(60°)
ω
=
V
R
0.2124
=
80
376.7
Assumptions
•
A level, coordinated turn: L cos φ = W holds the height and L sin φ turns the airplane.
•
The load factor n = L ÷ W is what the pilot feels, in g; g = 9.81 m/s².
•
The level part of the lift, W tan φ = W√(n² − 1), is the mass times V² ÷ R.
Step-by-step
WE KNOW
φ = 60°, V = 80 m/s
FIND
load factor (n), turn radius (R), turn rate (ω)
Step 1 · Find load factor (n)
Use n = 1 ÷ cos(φ)
Up
and
down:
L
cos
φ
=
W,
so
L
W
=
1
cos
φ
.
n
=
1
cos(60°)
n
=
1
0.5
n
=
2
Step 2 · Find turn radius (R)
Use R = V² ÷ (9.81 × tan(φ))
Sideways:
L
sin
φ
=
W
tan
φ
=
(
W
g
)
V
2
R
,
so
W
cancels.
R
=
80
2
9.81 × tan(60°)
R
=
80
2
9.81 × 1.732
R
=
80
2
16.99
R
=
6,400
16.99
R
=
376.7
m
Step 3 · Find turn rate (ω)
Use ω = V ÷ R
The speed round the circle over its radius.
ω
=
80
376.7
ω
=
0.2124
rad/s
Check
Put the numbers back in. Both sides should match.
2
=
1
cos(60°)
✓
376.7
=
80
2
9.81 × tan(60°)
✓
0.2124
=
80
376.7
✓
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