Tension in a rope holding a mass
Tension is the pulling force carried along a rope or cable, measured in newtons. For a hanging mass at rest the rope just holds its weight, so T = mg with g = 9.81 m/s². If the mass accelerates upward at a, the rope works harder and T = m(g + a). Accelerate it downward and a is negative, so tension drops; in free fall (a = −g) the rope carries nothing.
Tension in common setups
| Setup (10 kg) | Formula | Tension |
|---|---|---|
| Hanging, at rest | mg | 98.1 N |
| Pulled up at 5 m/s² | m(g + a) | 148.1 N |
| Falling freely | m(g − g) | 0 N |
| On a 30° frictionless incline | mg·sinθ | 49.1 N |
| On a 45° frictionless incline | mg·sinθ | 69.4 N |
On an incline only the along-slope share of gravity, mg·sinθ, pulls on the rope, so a steeper slope means more tension. An Atwood pulley with two masses gives T = 2m₁m₂g ÷ (m₁ + m₂).
Reading the result
- Units. 1 kgf = 9.80665 N and 1 lbf = 4.44822 N, so 98 N is about 10 kgf or 22 lbf. A quick check: at rest, newtons ≈ kilograms × 10.
- Acceleration matters most. Pulling a load up adds ma on top of its weight, which is why lift and crane cables are rated well above the static load.
- Ideal-rope assumptions. The formulas assume a massless, inextensible rope and a frictionless pulley. A heavy rope carries more tension near the top, and a real pulley makes the two sides unequal.
Common questions
How do you calculate the tension in a rope?
For a weight hanging still, tension equals mass times gravity: T = m x g. A 10 kg mass gives T = 10 x 9.81, about 98 newtons. If the mass accelerates upward at a, use T = m x (g + a).
Is tension the same as weight?
Only when the object is at rest or moving at constant speed, where T = mg. Accelerating the object upward raises the tension above its weight; letting it drop lowers the tension.
Can rope tension be negative?
No. A rope only pulls, never pushes, so tension is zero or positive. A negative result from a calculation means the rope has gone slack and the real tension is zero.
