Tension Calculator
Work out the pulling force in a rope, string or cable. Choose a situation, enter the mass and angles, and get each rope's tension with the free-body equations solved step by step.
Last updated:
same unit; m is mass 1
Earth 9.80665 · Moon 1.62 · Mars 3.71
0 for frictionless; wood on wood ≈ 0.3
0 = at rest or constant speed
How to use the Tension Calculator
- Pick the situation: one rope, two ropes, pulled along a floor, or two masses over a pulley.
- Enter the mass and its unit; keep gravity at 9.80665 m/s² for Earth.
- Add the angles (measured from the horizontal), friction coefficient or acceleration that the setup asks for.
- Press Calculate; results also update as you type.
- Check the steps and table to see how the tension changes with angle or acceleration.
Tension formulas for each setup
Tension is the pulling force carried along a rope, string, cable or chain. Every case here starts from a free-body diagram and Newton's laws, with the rope treated as light and unstretchable.
- One vertical rope:
T = m(g + a), with a positive when accelerating upward. At rest or constant speed, T = m·g. - Two ropes at angles θ₁, θ₂ from horizontal:
T₁ = W·cos θ₂ / sin(θ₁+θ₂),T₂ = W·cos θ₁ / sin(θ₁+θ₂). The horizontal parts cancel; the vertical parts add up to W. - Pulled along a floor at angle θ with friction μ:
T = m(a + μg) / (cos θ + μ·sin θ), with normal forceN = mg − T·sin θ. - Atwood machine:
T = 2m₁m₂g / (m₁ + m₂)anda = (m₁ − m₂)g / (m₁ + m₂).
Worked example: a 10 kg sign (W = 98.07 N) hangs from ropes at 30° and 60°. sin 90° = 1, so T₁ = 98.07 × cos 60° = 49.03 N and T₂ = 98.07 × cos 30° = 84.93 N. The steeper rope carries more of the load.
How rope angle multiplies the load
For a load shared equally by two ropes at the same angle from horizontal, each rope carries W / (2 sin θ):
| Angle from horizontal | Tension in each rope | Example: 100 kg load (981 N) |
|---|---|---|
| 90° (vertical) | 0.50 × W | 490 N |
| 60° | 0.58 × W | 566 N |
| 45° | 0.71 × W | 693 N |
| 30° | 1.00 × W | 981 N |
| 15° | 1.93 × W | 1,895 N |
| 5° | 5.74 × W | 5,626 N |
Tips for solving tension problems
- Measure angles consistently. This calculator uses the angle between the rope and the horizontal; if your problem gives the angle from the vertical, use 90° minus that value.
- A tight, nearly straight line (a clothesline, slackline or zip line) produces very large tensions even with a light load.
- Ropes can only pull. A negative result means the setup is impossible as drawn, for example the rope would have to push.
- For real lifting and rigging, use rated hardware and published working load limits with a safety factor; ideal physics values ignore shock loads, knots and wear.
Frequently asked questions
How do you calculate tension in a rope?
Draw the forces on the object and apply Newton's second law. For a mass hanging still from one vertical rope, the tension equals its weight, T = m·g. If the mass accelerates upward at a, T = m(g + a); if it accelerates downward, T = m(g − a).
What is the formula for tension in two ropes at angles?
With angles θ₁ and θ₂ measured from the horizontal, T₁ = W·cos θ₂ / sin(θ₁ + θ₂) and T₂ = W·cos θ₁ / sin(θ₁ + θ₂), where W = m·g. For equal angles θ this simplifies to T = W / (2·sin θ).
Why does tension increase when the rope angle gets flatter?
Only the vertical part of each rope's pull holds the weight up. As the rope nears horizontal, sin θ gets small, so the rope must pull much harder to give the same vertical support. At 30° from horizontal each of two ropes carries the full weight; at 10° it carries about 2.9 times the weight.
What is the tension in an Atwood machine?
For two masses on a light rope over a frictionless pulley, T = 2·m₁·m₂·g / (m₁ + m₂) and the acceleration is a = (m₁ − m₂)·g / (m₁ + m₂). The tension always lies between the two weights.
What angle should I pull a sled at to need the least force?
When friction coefficient μ applies, the tension needed is smallest when the rope angle above horizontal equals arctan(μ). With μ = 0.3 that is about 16.7°. Pulling slightly upward reduces the normal force and therefore the friction.
What units is tension measured in?
Tension is a force, so it is measured in newtons (N) in SI. 1 lbf = 4.44822 N and 1 kgf = 9.80665 N. A rope's working load limit is often quoted in kN, lbf or kg.
Does this account for rope weight or stretch?
No. Like standard physics problems, it assumes a light, inextensible rope and a frictionless, massless pulley. For real rigging, apply the manufacturer's working load limit and safety factors rather than relying on these idealized values.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.