Tension
Tension is the pulling force transmitted axially through a string, rope, cable, or rod when it is pulled taut by opposing forces at each end. Unlike compression, tension acts along the length of the medium and is directed away from the object on which it acts. Tension is central to the analysis of systems such as pulleys, pendulums, hanging masses, and suspension bridges.
Key Formula
T = m(g + a)
LaTeX: T = mg + ma
| Symbol | Meaning | Unit |
|---|---|---|
| T | Tension in the string/rope | N (Newton) |
| m | Mass of the hanging object | kg |
| g | Acceleration due to gravity | m/s² |
| a | Acceleration of the system (positive upward) | m/s² |
Worked Example
Problem
A 5 kg mass hangs from a rope and is accelerating upward at 2 m/s². What is the tension in the rope? (g = 9.8 m/s²)
Solution
Step 1 — Identify forces: Weight acts downward: W = mg = 5 × 9.8 = 49 N. Tension T acts upward. Step 2 — Apply Newton's second law (taking upward as positive): T − W = ma → T = m(g + a) = 5 × (9.8 + 2) = 5 × 11.8 = 59 N.
Answer
The tension in the rope is 59 N.
Tension in a rope for a 5 kg mass under different accelerations
| Condition | Acceleration (m/s²) | Direction | Tension (N) |
|---|---|---|---|
| At rest | 0 | — | 49.0 |
| Accelerating upward | 2 | Upward | 59.0 |
| Accelerating downward | 2 | Downward | 39.0 |
| Free fall | 9.8 | Downward | 0.0 (weightlessness) |
| Decelerating while descending | 3 | Upward | 64.0 |
Interactive Tools
Wikimedia Commons, CC BY-SA
From Latin "tensio" (a stretching), from "tendere" (to stretch or pull). The concept was analysed systematically by Isaac Newton in his "Principia Mathematica" (1687) when treating the forces in ropes and cables of mechanical systems.