What is the tension in the rope?

How it works

Each setup uses Newton's second law on light ropes that do not stretch, over pulleys with no mass and no friction:

• One rope holding a mass m, speeding up at a (up positive, 0 at rest or at a steady speed): T − mg = ma, so T = m × (g + a).
• Two ropes holding a mass m still, at θ₁ and θ₂ above level: level, T₁ cos θ₁ = T₂ cos θ₂; up, T₁ sin θ₁ + T₂ sin θ₂ = mg. So T₁ = mg cos θ₂ ÷ sin(θ₁ + θ₂) and T₂ = mg cos θ₁ ÷ sin(θ₁ + θ₂). With equal angles this is T = mg ÷ (2 sin θ), a tightrope or a sagging line.
• Pulley, two weights (an Atwood machine), m₁ on one side and m₂ on the other: a = (m₂ − m₁)g ÷ (m₁ + m₂), with m₂ moving down when a is positive, and T = 2m₁m₂g ÷ (m₁ + m₂).
• Table and pulley: m₁ on a frictionless table, pulled by m₂ hanging over the edge: a = m₂g ÷ (m₁ + m₂) and T = m₁m₂g ÷ (m₁ + m₂).

The page gives T in newtons and pounds-force (rope 1 as T and rope 2 as T₂ with two ropes), the weight m × g of the held mass (of m₂ with a pulley), and the acceleration of a pulley system.

Units (exact): 1 g = 0.001 kg; 1 t = 1,000 kg; 1 oz = 0.028349523125 kg; 1 lb = 0.45359237 kg; 1 short ton = 907.18474 kg; 1 lbf = 4.4482216152605 N. Angles are in degrees; g and a are in m/s² (9.80665 m/s² is standard gravity).

Rules

• Masses from 10⁻⁶ kg to 10¹² kg; g from 0.001 to 1,000 m/s²; acceleration from −1,000 to 1,000 m/s²; rope angles more than 0° and up to 90°. Limits are in SI units and include their ends (except 0° for an angle); a mass typed in another unit is converted first.
• One rope: if g + a is below 0 (the mass speeds down faster than gravity), the rope goes slack and there is no answer. Exactly at g + a = 0, T is 0.
• Two ropes both at 90° (straight up) have no answer: the split of the weight between them is not set.
• A result that is exactly 0 shows as 0. Any other result too large or too small to show as a number gives no answer.

Output format

The answer line reads T = a N (b lbf), or T₁ = a N, T₂ = b N with two ropes. It and every number show to 6 significant figures, rounded half up, with trailing zeros dropped and thousands commas, and below 10⁻⁶ or from 10¹⁵ up as a × 10ⁿ; negative numbers show with a minus sign (−). Typed values are read back as the exact decimals you typed, and every unit size is exact. sin and cos are exact where they are rational (0, ½ or 1, at 30°, 60° and 90°, and θ₁ + θ₂ at 30°, 90°, 150° or 180°), so those tensions are exact fractions rounded once; other angles use 64-bit floats.

Worked examples by hand

A rope at rest. 70 kg: T = 70 × 9.80665 = 686.4655 N, shown as T = 686.466 N (154.324 lbf).

Lifting with an acceleration. 75.0 kg at a = 1.20 m/s² up, g = 9.80 m/s²: T = 75 × (9.8 + 1.2) = 825 N, as the scale reading of OpenStax Example 6.3.

Two wires. OpenStax Example 6.1: a 15.0 kg traffic light (w = 15 × 9.8 = 147 N) on wires at 30° and 45°: T₁ = 147 × cos 45° ÷ sin 75° = 147 × 0.707107 ÷ 0.965926 = 107.611 N and T₂ = 147 × cos 30° ÷ sin 75° = 131.797 N (OpenStax: 108 N and 132 N).

A tightrope. OpenStax Example 5.13: 70.0 kg, the wire at 5.0° each side, g = 9.80 m/s²: T = 686 × cos 5° ÷ sin 10° = 686 ÷ (2 × sin 5°) = 3,935.48 N (OpenStax rounds to 3,930 N).

An Atwood machine. OpenStax Example 6.5: m₁ = 2.00 kg, m₂ = 4.00 kg, g = 9.8 m/s²: a = (4 − 2) × 9.8 ÷ 6 = 3.26667 m/s² and T = 2 × 2 × 4 × 9.8 ÷ 6 = 26.1333 N.

A block on a table. m₁ = 5.00 kg on the table, m₂ = 3.00 kg hanging, g = 9.8 m/s²: a = 3 × 9.8 ÷ 8 = 3.675 m/s² and T = 5 × 3 × 9.8 ÷ 8 = 18.375 N.