Tension
Tension is the pulling force transmitted along a string, rope, cable, or any similar connector when it is taut. Unlike a push (compression), a rope can only pull — it cannot push. When you hang a bag from a hook using a cord, the cord is under tension: every tiny segment of the cord pulls on the segment next to it, transmitting the weight of the bag upward to the hook. Tension is not stored in one place; it is distributed throughout the entire length of the connector, which is why a guitar string vibrates along its whole length when plucked. In mechanics problems, we almost always idealise the rope as massless and inextensible. Massless means the rope itself has no weight to support, so it does not change the tension along its length. Inextensible means it does not stretch, so every point along the rope accelerates identically. These idealisations simplify calculations enormously without sacrificing much accuracy in most real situations — a light nylon cord holding a textbook behaves almost exactly like an ideal massless string. When these conditions are not met (e.g. a heavy steel cable), the analysis becomes significantly more complex. Tension always acts along the line of the rope and points away from the object being considered — it pulls rather than pushes. On a free-body diagram, tension arrows point from the object toward the rope. Recognising the correct direction of tension is essential for applying Newton's second law correctly to each object in a connected system.
- The rope as a force transmitter
- Drawing free-body diagrams for tension
- Applying Newton's second law
- Tension in angled ropes
The rope as a force transmitter
Think of a rope as a chain of links. Each link pulls its neighbour. The force each link exerts on the next is the tension at that point. For a massless rope, Newton's second law applied to any tiny segment (mass ≈ 0) demands that the net force on it is zero, so the tension is the same throughout the rope. This is why a single value T can describe the whole rope.
Drawing free-body diagrams for tension
Isolate each object in the system. For each surface where a rope is attached, draw a tension arrow pointing along the rope, away from the object. Do this separately for every object — the rope transmits the same magnitude T to both ends, but the directions are opposite. For example, in an Atwood machine (two masses over a pulley), the rope pulls mass A upward and mass B upward, yet the two masses may accelerate in opposite directions.
Applying Newton's second law
Once free-body diagrams are drawn, apply ΣF = ma to each object along the direction of motion. For a hanging mass in equilibrium: T − mg = 0, so T = mg. For an accelerating system, T will be less than mg for a descending mass and greater than mg for an ascending one. Always define a consistent positive direction before writing equations — inconsistency here is the most common source of errors.
Tension in angled ropes
When a rope makes an angle θ with the horizontal (e.g. pulling a sled along the ground at an angle), only the component of tension along the direction of motion does work on the object. Resolve T into horizontal (T cos θ) and vertical (T sin θ) components. The vertical component partially reduces the normal force, which in turn affects friction. Always resolve forces before summing them — never add magnitudes from different directions directly.
The Key Equation
- T — Tension (Newton (N)): The pulling force exerted by the rope on the object, directed upward along the rope.
- m — Mass (kilogram (kg)): The mass of the hanging object.
- g — Gravitational field strength (m s⁻²): Acceleration due to gravity, approximately 9.81 m s⁻² near Earth's surface. The product mg is the object's weight.
- a — Acceleration (m s⁻²): The acceleration of the object. Positive upward in this convention. Zero if the system is in equilibrium.
Worked Example
Problem. Two blocks are connected by a light, inextensible string over a frictionless pulley (an Atwood machine). Block A has mass 3.0 kg and Block B has mass 5.0 kg. The system is released from rest. Find the acceleration of the system and the tension in the string. Take g = 9.81 m s⁻².
Answer. The acceleration of the system is 2.45 m s⁻² and the tension in the string is 36.8 N.
Origins · 1 minIsaac Newton and how this was discovered
Newton formulated the three laws of motion in his landmark 1687 work Philosophiæ Naturalis Principia Mathematica. His second law, F = ma, is the foundation upon which all tension analysis is built. By treating forces as vectors and applying his laws to connected bodies, Newton provided the complete mathematical framework that allows us to calculate tension in any mechanical system. Newton reportedly developed his laws of motion and law of universal gravitation during 1665–1666, when Cambridge University was closed due to the Great Plague — arguably the most productive enforced break in scientific history.
Related Concepts
- Newton's Second Law — Tension is analysed by applying Newton's second law (ΣF = ma) to each object in the system — it is the primary tool for calculating tension values.
- Normal Force — Like tension, the normal force is a contact force that appears in free-body diagrams; understanding both is essential for solving connected-body problems.
- Friction — Many tension problems involve surfaces with friction — the tension required to accelerate a block depends directly on the frictional force opposing its motion.
- Free-Body Diagrams — Correctly drawing tension arrows on free-body diagrams is the essential first step in solving any tension problem — direction errors lead to incorrect equations.