SME
Statics
Force vectors and equilibrium
Use this for “A 500 N sign hangs from two cables at 30° and 60° above level. Find each tension.”
Diagram
T₁ = 250 N at 30° from the horizontal: (−216.5, 125) N.
T₂ = 433 N at 60° from the horizontal: (216.5, 375) N.
W = 500 N at 90° from the horizontal: (0, −500) N.
ΣFₓ = 0 N and ΣFᵧ = 0 N.
The forces balance: tip to tail, the polygon closes.
This is an example. Type your own values to start a new problem.
T₁ (N)
=
W (N)
×
cos(
°)
sin(
°
+
°)
T₂ (N)
=
W (N)
×
cos(
°)
sin(
°
+
°)
Level balance: T₁ cos θ₁ = T₂ cos θ₂. Upright balance: T₁ sin θ₁ + T₂ sin θ₂ = W. Solving the two together gives each tension over sin(θ₁ + θ₂).
Formulas
T₁
=
W
×
cos(
θ₂
°)
sin(
θ₁
° +
θ₂
°)
250
=
500
×
cos(60°)
sin(30° + 60°)
T₂
=
W
×
cos(
θ₁
°)
sin(
θ₁
° +
θ₂
°)
433.0127
=
500
×
cos(30°)
sin(30° + 60°)
T₁
×
cos(
θ₁
°)
=
T₂
×
cos(
θ₂
°)
250 × cos(30°) = 433.0127 × cos(60°)
Assumptions
•
The knot is a particle: the forces on it balance, ΣFₓ = 0 and ΣFᵧ = 0, and moments don’t enter.
•
Cables only pull, along their length, so T₁ and T₂ are never negative.
•
Each angle is measured above the horizontal, cable 1 to the left and cable 2 to the right; + x right, + y up.
Step-by-step
WE KNOW
W = 500 N, θ₁ = 30°, θ₂ = 60°
FIND
left cable tension (T₁), right cable tension (T₂)
Step 1 · Find left cable tension (T₁)
Use T₁ = W × cos(θ₂°) ÷ sin(θ₁° + θ₂°)
Solve
ΣFₓ
=
0
and
ΣFᵧ
=
0
together:
the
left
cable
carries
W
times
the
cosine
of
the
other
cable’s
angle,
over
sin(
θ₁
+
θ₂
).
T₁
=
500
×
cos(60°)
sin(30° + 60°)
T₁
=
500
×
cos(60°)
sin(90°)
T₁
=
500
×
0.5
1
T₁
=
250
N
Step 2 · Find right cable tension (T₂)
Use T₂ = W × cos(θ₁°) ÷ sin(θ₁° + θ₂°)
Solve
ΣFₓ
=
0
and
ΣFᵧ
=
0
together:
the
right
cable
carries
W
times
the
cosine
of
the
other
cable’s
angle,
over
sin(
θ₁
+
θ₂
).
T₂
=
500
×
cos(30°)
sin(30° + 60°)
T₂
=
500
×
cos(30°)
sin(90°)
T₂
=
500
×
0.866
1
T₂
=
433.0127
N
Check
Put the numbers back in. Both sides should match.
250
=
500
×
cos(60°)
sin(30° + 60°)
✓
433.0127
=
500
×
cos(30°)
sin(30° + 60°)
✓
250 × cos(30°) = 433.0127 × cos(60°) ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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