SME
Grade 12 Math · Problem type
The resultant of two forces
Use this for “Forces of 30 N at 0° and 40 N at 60° pull on a ring. Find the resultant.”
Diagram
F₁: 30 N at 0°, so F₁ = ⟨30, 0⟩.
F₂: 40 N at 60°, so F₂ = ⟨20, 34.64⟩.
Parallelogram: F = ⟨50, 34.64⟩
|F| = 60.83 N at 34.72°.
This is an example. Type your own values to start a new problem.
Fₓ (N)
=
F₁ (N)
+
F₂ (N)
cos
a
°
Fᵧ (N)
=
F₂ (N)
sin
a
°
F
Resultant
calculated
φ
Direction of the resultant
calculated · °
Formulas
Fₓ
=
F₁
+
F₂
×
cos(
a
°)
50 = 30 + 40 × cos(60°)
Fᵧ
=
F₂
×
sin(
a
°)
34.641 = 40 × sin(60°)
F
=
√
Fₓ² + Fᵧ²
60.8276
=
√
50
2
+
34.641
2
tan
φ
=
Fᵧ
Fₓ
,
φ
in
the
quadrant
of
(Fₓ,
Fᵧ)
tan
34.715
=
34.641
50,
34.715
in
the
quadrant
of
(50,
34.641)
Assumptions
•
F₁ points along the positive x-axis; F₂ makes angle a with it.
•
Add the components: the resultant is the diagonal of the parallelogram the forces make.
•
Its direction φ is measured from F₁, counterclockwise.
Step-by-step
WE KNOW
F₁ = 30 N, F₂ = 40 N, a = 60°
FIND
x-component of the resultant (Fₓ), y-component of the resultant (Fᵧ), resultant (F), direction of the resultant (φ)
Step 1 · Find x-component of the resultant (Fₓ)
Use Fₓ = F₁ + F₂ × cos(a°)
F₁
is
all
x;
F₂
adds
its
x-component,
F₂
cos
a
.
Fₓ = 30 + 40 × cos(60°)
Fₓ = 30 + 40 × 0.5
Fₓ = 30 + 20
Fₓ = 50 N
Step 2 · Find y-component of the resultant (Fᵧ)
Use Fᵧ = F₂ × sin(a°)
Only
F₂
has
a
y-component,
F₂
sin
a
.
Fᵧ = 40 × sin(60°)
Fᵧ = 40 × 0.866
Fᵧ = 34.641 N
Step 3 · Find resultant (F)
Use F = √(Fₓ² + Fᵧ²)
The resultant is the hypotenuse of its components.
F
=
√
50
2
+
34.641
2
F
=
√
2,500 + 1,200
F
=
√
3
,700
F
=
60.8276
N
Step 4 · Find direction of the resultant (φ)
Use tan φ = Fᵧ ÷ Fₓ, φ in the quadrant of (Fₓ, Fᵧ)
tan⁻¹ gives an angle from −90° to 90°; add 180° when the resultant points left, 360° when it points down and right.
φ
=
tan⁻¹(
Fᵧ
Fₓ
)
φ
=
tan⁻¹(
34.641
50
)
φ
=
34.715°
Check
Put the numbers back in. Both sides should match.
50 = 30 + 40 × cos(60°) ✓
34.641 ≈ 40 × sin(60°) ✓
60.8276
≈
√
50
2
+
34.641
2
✓
tan(34.715°)
=
34.641
50
✓
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