SME
Grade 11 Science · Problem type
Adding two forces
Use this for “Two dogs pull a 25 kg sled: 60 N east and 80 N north. What is the net force, which way, and the acceleration?”
Diagram
F₁: 60 N at 0°, so F₁ = ⟨60, 0⟩.
F₂: 80 N at 90°, so F₂ = ⟨0, 80⟩.
Parallelogram: F = ⟨60, 80⟩
|F| = 100 N at 53.13°.
Mass: 25 kg
Acceleration: 4 m/s²
This is an example. Type your own values to start a new problem.
F₁
First force
example
F₂
Second force
example
θ
Angle of F₂
example · °
Fₓ
Net force east
worked out (not typed)
Fᵧ
Net force north
worked out (not typed)
F
Net force
calculated
φ
Direction, from east
worked out (not typed) · °
m
Mass
example
a
Acceleration
calculated · m/s²
Formulas
Fₓ
=
F₁
+
F₂
×
cos(
θ
)
60 = 60 + 80 × cos(90°)
Fᵧ
=
F₂
×
sin(
θ
)
80 = 80 × sin(90°)
F
=
√
Fₓ² +
Fᵧ
2
100
=
√
60
2
+
80
2
tan(
φ
)
=
Fᵧ/Fₓ
tan(53.1°)
=
80
60
a
=
F
m
4
=
100
25
Assumptions
•
F₁ pulls east; F₂ pulls at the angle θ turned counterclockwise from east (90° north, 270° south). No other force acts sideways.
•
The net force is their vector sum: add the east parts and the north parts.
•
The acceleration points the way the net force does: a = F ÷ m.
Step-by-step
WE KNOW
F₁ = 60 N, F₂ = 80 N, θ = 90°, m = 25 kg
FIND
net force east (Fₓ), net force north (Fᵧ), net force (F), direction, from east (φ), acceleration (a)
Step 1 · Find net force east (Fₓ)
Use Fₓ = F₁ + F₂ × cos(θ)
Add
the
east
parts:
all
of
F₁
,
and
F₂
times
cos
θ
.
Fₓ = 60 + 80 × cos(90°)
Fₓ = 60 + 80 × 0
Fₓ = 60 N
Step 2 · Find net force north (Fᵧ)
Use Fᵧ = F₂ × sin(θ)
Only
F₂
pulls
north:
F₂
times
sin
θ
.
Fᵧ = 80 × sin(90°)
Fᵧ = 80 N
Step 3 · Find net force (F)
Use F = √(Fₓ² + Fᵧ²)
The parts are at right angles: add them by Pythagoras.
F
=
√
60
2
+
80
2
F
=
√
3,600 + 6,400
F
=
√
10
,000
F
=
100
N
Step 4 · Find direction, from east (φ)
Use tan(φ) = Fᵧ/Fₓ
The direction from east: the angle whose tangent is north over east, in the right quarter.
φ
=
arctan(Fᵧ/Fₓ)
φ
=
arctan(
80
60
)
φ
=
53.1°
Step 5 · Find acceleration (a)
Use a = F/m
Newton’s second law: the net force over the mass.
a
=
100
25
a
=
4
m/
s
2
Check
Put the numbers back in. Both sides should match.
60 = 60 + 80 × cos(90°) ✓
80 = 80 × sin(90°) ✓
100
=
√
60
2
+
80
2
✓
tan(53.13°)
=
80
60
✓
4
=
100
25
✓
Topic check
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