SME
Grade 11 Science · Problem type
Pulling by a rope at an angle
Use this for “A 10 kg sled is pulled with 50 N by a rope 30° above level; μₖ = 0.1. Find the normal force and the acceleration.”
Diagram
Weight = m × g = 10 × 9.8 = 98 N
Normal force = Weight − Tension sin 30° = 98 − 50 × 0.5 = 73 N
f = μₖ × normal force = 0.1 × 73 = 7.3 N
Net force = Tension cos 30° − f = 43.3 − 7.3 = 36 N
a = Net force/m = 36/10 = 3.6 m/s²
This is an example. Type your own values to start a new problem.
m
Mass
example
T
Tension in the rope
example
α
Angle of the rope
example · °
μₖ
Coefficient of kinetic friction
example
W
Weight
calculated
FN
Normal force
calculated
f
Friction
calculated
Fₙₑₜ
Net force
calculated
a
Acceleration
calculated · m/s²
Formulas
W
=
m
×
9.8
98 = 10 × 9.8
FN
=
W
−
T
×
sin(
α
)
73 = 98 − 50 × sin(30°)
f
=
μₖ
×
FN
7.3 = 0.1 × 73
Fₙₑₜ
=
T
×
cos(
α
)
−
f
36 = 50 × cos(30°) − 7.3
a
=
Fₙₑₜ/
m
3.6
=
36
10
Assumptions
•
The rope’s tension has a part across, T cos α, and a part up, T sin α.
•
The part up lifts a little, so the ground pushes up less: FN = W − T sin α.
•
Less normal force means less friction; the sled is already sliding, and a < 0 means it slows.
Step-by-step
WE KNOW
m = 10 kg, T = 50 N, α = 30°, μₖ = 0.1
FIND
weight (W), normal force (FN), friction (f), net force (Fₙₑₜ), acceleration (a)
Step 1 · Find weight (W)
Use W = m × 9.8
Earth pulls each kilogram with 9.8 N.
W
=
10
×
9.8
W
=
98
N
Step 2 · Find normal force (FN)
Use FN = W − T × sin(α)
The
rope
lifts
T
sin
α
of
the
weight;
the
ground
holds
the
rest.
FN = 98 − 50 × sin(30°)
FN = 98 − 50 × 0.5
FN = 98 − 25
FN = 73 N
Step 3 · Find friction (f)
Use f = μₖ × FN
Friction is μₖ times the normal force pressing the surfaces together.
f
=
0.1
×
73
f
=
7.3
N
Step 4 · Find net force (Fₙₑₜ)
Use Fₙₑₜ = T × cos(α) − f
The rope’s pull across, less friction.
Fₙₑₜ = 50 × cos(30°) − 7.3
Fₙₑₜ = 50 × 0.866 − 7.3
Fₙₑₜ = 43.3 − 7.3
Fₙₑₜ = 36 N
Step 5 · Find acceleration (a)
Use a = Fₙₑₜ/m
Newton’s second law: the net force over the mass.
a
=
36
10
a
=
3.6
m/
s
2
Check
Put the numbers back in. Both sides should match.
98 = 10 × 9.8 ✓
73 = 98 − 50 × sin(30°) ✓
7.3 = 0.1 × 73 ✓
36 ≈ 50 × cos(30°) − 7.3 ✓
3.6
=
36
10
✓
Topic check
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