Rope Tension Calculator
The tension in a rope for the four cases that come up: hanging, in a lift, over a pulley, and shared between two ropes.
Results
What this tool does
Tension is whatever the rope has to pull with, and it is only equal to the weight in the simplest case of all — something hanging still. Put the same mass in a lift that is accelerating upwards and the rope has to do more; hang it from two ropes and the answer depends on their angles far more than most people expect; run it over a pulley against another mass and the tension drops below either weight. This page does all four, so you can see how much the arrangement matters.
Formula
at rest: T = mg · accelerating: T = m(g + a) · two ropes: T₁ = W cos θ₂ ÷ sin(θ₁ + θ₂)
Variables
| Symbol | Meaning | Unit |
|---|---|---|
mode | The arrangement | — |
mass | Mass | kg |
accel | Acceleration (upwards positive) | m/s² |
mass_2 | The other mass | kg |
angle_1 | First rope's angle from horizontal | ° |
angle_2 | Second rope's angle from horizontal | ° |
gravity | Gravitational acceleration | m/s² |
T1 | Tension in the rope | N |
T2 | Tension in the second rope | N |
AC | Acceleration | m/s² |
WT | Weight | N |
TR | Tension compared to the weight | x |
TS | Upward pull, which must equal the weight | N |
Worked example
- The arrangementhanging
- Mass10 kg
- Acceleration (upwards positive)0 m/s²
- The other mass6 kg
- First rope's angle from horizontal45 °
- Second rope's angle from horizontal45 °
- Gravitational acceleration9.80665 m/s²
- Tension in the rope98.0665 N
- Tension in the second rope0.0000 N
- Acceleration0.0000 m/s²
- Weight98.0665 N
- Tension compared to the weight1.0000 x
- Upward pull, which must equal the weight98.0665 N
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
Frequently asked questions
Why does spreading two ropes wider increase the tension?
Because only the vertical part of each rope's pull holds the weight, and the flatter the rope, the smaller that part is. At forty-five degrees each rope carries about seventy per cent of the weight. At ten degrees from horizontal each carries nearly three times the weight, and almost all of that is spent pulling against the other rope rather than holding anything up. This is why a washing line pulled truly straight will snap under a light load, and why rigging is never set horizontal.
Why is the tension in a pulley system less than either weight?
Because the system is accelerating, and an accelerating mass does not need its full weight supported. With six kilograms against four, the heavier side falls and the lighter side rises; the rope only has to carry enough to explain the motion, which works out at 48 newtons rather than the 60 the heavier mass weighs. If you locked the pulley, the tension would jump straight back to the full weight of whichever side you were holding.