A climber hangs from a rope. A tow truck pulls a car. A pendulum bob swings on a string. In each case, a pulling force acts along the rope or string — the tension force. Tension is a pulling force transmitted through a flexible connector (rope, string, cable, chain), always directed away from the object and along the connector toward the point of attachment. It is one of the most common forces in mechanics problems and one of the most frequently confused — particularly around what tension actually means, and how it varies (or doesn't) along an ideal string.
Tension is the pulling force transmitted along a rope, string, cable, or rod under traction. It acts along the connector, directed away from the object being considered and toward the point of attachment. In an ideal (massless, inextensible) string, tension is the same throughout. Tension is a contact force — it can only pull, never push (unlike a rod, which can also push in compression).
Tension in a Hanging Object
The simplest case: a mass m hanging stationary from a string attached to the ceiling. Applying Newton's second law vertically (taking upward positive, a = 0):
The tension equals the weight. This changes immediately if the object is accelerating:
Object accelerating upward at a:
Object accelerating downward at a:
In free fall (a = g): T = m(g − g) = 0 — the string goes slack.
The Atwood Machine
The Atwood machine is a classic problem: two masses m₁ and m₂ (m₁ > m₂) connected by a string over a frictionless, massless pulley. The heavier mass descends, accelerating both masses. Applying Newton's second law to each mass (treating the system as a whole first):
The tension (same throughout the ideal string) is found by applying Newton's law to either mass:
Worked example: Atwood machine
m₁ = 5 kg, m₂ = 3 kg.
Check: For m₁: mg − T = m₁a → 49 − 36.75 = 12.25 = 5 × 2.45 ✓
Check: For m₂: T − mg = m₂a → 36.75 − 29.4 = 7.35 = 3 × 2.45 ✓
Tension on an Inclined Plane
A mass m on a frictionless incline of angle θ is held by a string parallel to the slope. Forces along the slope (taking up the slope as positive):
If the mass accelerates up the slope at a:
Worked example: Mass on slope
A 4 kg mass on a 30° frictionless slope is pulled up the slope with acceleration 2 m/s². Find the tension.
Tension in a String at an Angle
When a string pulls at angle θ to the horizontal, the tension has both horizontal and vertical components:
Worked example: Two-string suspension
A 10 kg traffic light hangs from two strings making 30° and 60° to the horizontal respectively. Find the tension in each string.
Vertical equilibrium: T₁ sin30° + T₂ sin60° = mg = 98 N
Horizontal equilibrium: T₁ cos30° = T₂ cos60°
From horizontal: T₁ × 0.866 = T₂ × 0.5 → T₁ = 0.577 T₂
Substituting: 0.577T₂ × 0.5 + T₂ × 0.866 = 98 → 0.289T₂ + 0.866T₂ = 98 → T₂ = 84.5 N
T₁ = 0.577 × 84.5 = 48.8 N
An ideal (massless, inextensible) string has tension that is uniform throughout — the same at every point. This means if you pull one end with force T, the other end also exerts force T. A real string has mass, so tension varies along its length. A massless pulley changes the direction of tension but not its magnitude; a massive pulley also changes the magnitude. These idealisations simplify most introductory mechanics problems.
Tension vs Compression
Tension and compression are opposite types of internal force in a structural element:
Tension: the element is being stretched — it pulls both ends toward its centre. Ropes, strings, cables, and tendons work in tension. They cannot support compression (they would go slack).
Compression: the element is being compressed — it pushes both ends away from its centre. Columns, pillars, and bone work in compression. They cannot support tension (they would buckle or fracture).
Structural engineering involves designing systems where each element carries the appropriate type of force — cables in tension, columns in compression. Bridges combine both: suspension bridges use cables in tension to support a deck in a mix of compression and bending.
Frequently Asked Questions
What is tension force?
Tension is the pulling force transmitted through a rope, string, or cable under traction. It acts along the connector, pulling the connected object toward the attachment point. In an ideal massless string, tension is the same everywhere along the string. Tension can only pull — ropes and strings cannot push (they go slack instead).
How do you calculate tension in a rope?
Draw a free-body diagram of the object, identify all forces, and apply Newton's second law (F_net = ma) in the relevant direction. For a hanging mass at rest: T = mg. For a mass accelerating upward at a: T = m(g + a). For a mass on a frictionless incline: T = mg sinθ (if stationary) or T = m(a + g sinθ) if accelerating up the slope.
Is tension the same throughout a rope?
In an ideal massless, inextensible rope — yes. Tension is uniform throughout. In a real rope with mass, tension varies: it is greatest at the top (supporting its own weight plus the object below) and least at the bottom. For most introductory physics problems, the rope is treated as massless and tension is assumed uniform.
What is an Atwood machine?
An Atwood machine consists of two masses connected by a string over a frictionless, massless pulley. It is used to study Newton's second law. The acceleration is a = (m₁ − m₂)g/(m₁ + m₂) and the tension is T = 2m₁m₂g/(m₁ + m₂). If the masses are equal, a = 0 and T = mg (just supporting each mass).
Can tension be negative?
No — tension is always positive or zero. A rope can only pull, not push. If your calculation gives a negative tension, it means the rope would need to push, which is physically impossible — the rope goes slack and T = 0. This often signals an error in the direction assignment or that the assumed direction of motion is wrong.
Share this article
Written by
Dr. James CarterPhysicist and educator with 15+ years teaching classical mechanics and thermodynamics at the university level. Former MIT OpenCourseWare contributor.
View all articles by this author →
