SME
Statics · Problem type
Belt or rope on a drum
Use this for “A rope passes half way round a fixed post (μ = 0.3) with 100 N on the slack side. What pull on the other end makes it slip?”
Diagram
T₂
=
T₁
e
μβ
T₂
=
100
×
e
0.3 × 3.142
=
256.6
N
β is in radians in the formula; the rope is about to slip toward T₂.
This is an example. Type your own values to start a new problem.
T₂ (N)
=
T₁ (N)
×
e
×
°
×
π
÷
180
Formulas
T₂
=
T₁
×
e
μ
×
β
× π ÷ 180
256.6332
=
100
×
e
0.3 × 180 × π ÷ 180
Assumptions
•
The rope is about to slip on the fixed drum, toward the tight side T₂.
•
β is the angle of contact; the formula takes it in radians (β° × π ÷ 180).
•
A light, flat rope or belt (no V-groove); μ is the static coefficient.
Step-by-step
WE KNOW
T₁ = 100 N, μ = 0.3, β = 180°
FIND
tight-side tension (T₂)
Step 1 · Find tight-side tension (T₂)
Use T₂ = T₁ × eμ × β × π ÷ 180
Each bit of contact adds friction in proportion to the tension there, so the tension grows exponentially round the drum.
T₂
=
100
×
e
0.3 × 180 × π ÷ 180
β
=
180
×
π
180
=
3.1416
rad
T₂
=
100
×
e
0.3 × 3.1416
T₂
=
100
×
2.5663
T₂
=
256.6332
N
Check
Put the numbers back in. Both sides should match.
256.6332
=
100
×
e
0.3 × 180 × π ÷ 180
✓
Topic check
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