Tension Calculator

Tension Calculator finds the tension in 1 or 2 ropes holding a load at any angle, in ropes pulling 2 or 3 objects, and in a string over a pulley, with a diagram and the full steps.

Tension Calculator for Ropes, Strings and Pulleys

Tension is the pulling force carried along a rope, string, cable or wire when it is pulled tight. OpenStax University Physics calls it a force along the length of a flexible connector. A rope can only pull, never push, and it pulls equally on whatever is tied to each end.

The Tension Calculator covers five rope setups from the Rope Setup menu. A load can hang from 1 rope or from 2 ropes at different angles. A force can pull 2 or 3 objects joined in a row. The last setting is two masses hanging over a pulley.

Masses go in as kilograms, grams, pounds or slugs. Results come out in newtons, kilonewtons, pounds-force or kilograms-force, chosen under Result Unit and Gravity. Gravity starts at 9.80665 m/s² and can be changed there for another planet or a rounded textbook value.

With the starting entries, a 10 kg load hangs from two ropes at 60° and 40° to the ceiling. The main result reads T₁ = 76.28 N, T₂ = 49.79 N. A Setup Diagram under the result cards of the Tension Calculator draws the ropes, the angles and the load.

Tension Formula for Each Rope Setup

There is no single formula for tension. Tension is whatever force the rope must supply to make Newton’s second law work for the object on its end. The tension calculator formula therefore changes with the setup.

To solve tension problems by hand, draw the forces on each object and write the second law along each direction. Then solve those equations for T. The Step-by-Step Working panel in the Tension Calculator writes out those lines with your own numbers.

Is Tension Equal to mg?

Tension equals m × g only for a load that hangs from one vertical rope and is not accelerating. That covers a load at rest and a load moving up or down at a steady speed. A 5 kg mass on a rope gives 5 × 9.80665 = 49.03 N.

When the load accelerates, the rope has to supply the net force as well. In the Tension Calculator, upward acceleration is entered as a positive number and downward as a negative one.

$$T = m (g + a)$$

A 10 kg load lifted with an acceleration of 2 m/s² needs 10 × (9.80665 + 2) = 118.1 N. Lowered with a downward acceleration of 2 m/s², the same load needs only 78.07 N. The second result card shows this as a tension-to-weight ratio, 1.204 and 0.796.

The same formula sets the maximum tension in a lift. The peak comes at the largest upward acceleration, usually as the load starts to rise or is stopped on the way down. That peak value is the one to compare with the rope’s rated working load.

If the downward acceleration reaches g, the tension falls to zero and the tool reports Rope Goes Slack. The load is then in free fall. The Normal Force Calculator handles the matching case of a floor in an accelerating elevator.

How to Calculate Tension at an Angle

A load hanging from two ropes is the standard tension formula with angle. Each rope pulls partly upward and partly sideways. The sideways parts must cancel, and the upward parts must add up to the weight W = m × g.

$$T_{1} \cos\alpha = T_{2} \cos\beta$$

$$T_{1} \sin\alpha + T_{2} \sin\beta = W$$

Solving the pair gives one formula for each rope. The angles α and β are measured between each rope and the ceiling, so 90° means the rope hangs straight down and 0° means it is level.

$$T_{1} = \frac{W \cos\beta}{\sin(\alpha + \beta)}$$

$$T_{2} = \frac{W \cos\alpha}{\sin(\alpha + \beta)}$$

For 10 kg at 60° and 40°, W is 98.07 N and sin(100°) is 0.9848. T₁ is 98.07 × 0.7660 ÷ 0.9848 = 76.28 N. T₂ is 98.07 × 0.5 ÷ 0.9848 = 49.79 N. The steeper rope carries more, here 67.36% of the weight.

OpenStax works the same method for a 15 kg traffic light on wires at 30° and 45°, and gets 108 N and 132 N. Entering those values in the Tension Calculator returns 107.7 N and 131.9 N, with the small gap coming from g = 9.80665 instead of 9.80.

When both ropes sit at the same angle θ, the two tensions are equal and the formula shortens. A 10 kg box on two ropes at 60° has 56.62 N in each, which is more than half its weight.

$$T = \frac{W}{2 \sin\theta}$$

These rows come from the Tension at Other Rope Angles panel of the Tension Calculator. At 30° each rope already carries the full weight, and below that the tension climbs fast. The third result card warns of this by showing the peak tension if both of your angles were halved.

The fourth card gives what each ceiling anchor must resist. For the 60° and 40° example that is 38.14 N sideways at each anchor, with 66.06 N and 32 N pulling down on them. The Net Force Calculator can confirm that the two tensions and the weight add to zero.

Line Tension and Horizontal Cable Tension

A nearly level line is the extreme case of the same formula. OpenStax uses a 70 kg tightrope walker on a wire that sags only 5° on each side. The Tension Calculator gives 3,938 N in the wire, which is 5.7 times the walker’s weight.

Slackliners calculate line tension from the sag instead of the angle. Balance Community gives a short formula for a person of weight W standing at the middle of a line of length L with sag S.

$$T \approx \frac{W L}{4 S}$$

A 75 kg person on a 20 m line with 1 m of sag gives 735.5 × 20 ÷ 4 = 3,677 N. In the tool, the same line is two ropes at tan⁻¹(2S ÷ L) = 5.71°, which returns about 3,696 N. The two agree closely as long as the sag is small next to the length.

A horizontal cable that carries only its own weight, such as an overhead electrical conductor, follows a related rule. With a weight per unit length w, a span L and a mid-span sag d, the tension is close to the value below. Pulling a cable flatter always costs tension.

$$T \approx \frac{w L^{2}}{8 d}$$

How to Calculate Tension with a Pulling Force

The two pulling settings model objects tied in a row on a frictionless surface. Rope 1 pulls the first object with a known force at an angle θ above the level. Rope 2 joins objects 1 and 2, and rope 3 joins objects 2 and 3.

Only the forward part of the pull, T₁ × cos θ, moves the objects. All of them share one acceleration, which is that forward part divided by the total mass. With two objects, m₃ is left out.

$$a = \frac{T_{1} \cos\theta}{m_{1} + m_{2} + m_{3}}$$

Each connecting rope then pulls only the masses behind it. With 24 N at 60° on a 3 kg and a 2 kg object, the forward pull is 12 N and the acceleration is 2.4 m/s². Rope 2 pulls the 2 kg object alone, so T₂ = 2 × 2.4 = 4.8 N.

Adding a third object of 4 kg slows everything to 1.333 m/s². Rope 2 now pulls 6 kg and carries 8 N, while rope 3 pulls 4 kg and carries 5.333 N. The rope nearest the pull always has the highest tension.

The fourth card of the Tension Calculator shows what the angle costs. Pulled level, the same 24 N would put 9.6 N in rope 2, so 50% of the pull is lost at 60°. Friction is left out of this model, and the Friction Calculator gives the force to subtract when a surface is rough.

Tension in a String over a Pulley

Two masses on one string over a pulley is the Atwood machine, a standard A level and first-year physics problem. The string is taken as light and the pulley as free to turn, so the tension is the same on both sides.

$$T = \frac{2 m_{1} m_{2} g}{m_{1} + m_{2}}$$

$$a = \frac{|m_{2} – m_{1}| \, g}{m_{1} + m_{2}}$$

For the 3 kg and 2 kg that the Tension Calculator starts with, the tension is 2 × 3 × 2 × 9.80665 ÷ 5 = 23.54 N. The acceleration is 1.961 m/s², which is 20% of free fall. The heavier side moves down and the lighter side moves up at the same rate.

The tension always lies between the two weights, here 19.61 N and 29.42 N. The third card adds the load on the pulley axle, 47.07 N, which is twice the tension. That is less than the combined weight of 49.03 N because the masses are accelerating.

The acceleration found here can be carried into other problems, such as how long a mass takes to reach the floor. The Newton’s Second Law Calculator and the Acceleration Calculator take it further to speeds and times.

Tension Units and How Tension Is Measured

Tension is a force, so the SI unit of tension is the newton. One newton is the force that accelerates 1 kg at 1 m/s². Tension is also quoted in pounds-force in the US and in kilograms-force on many gauges, where 1 kgf is 9.80665 N.

The first result card lists the answer in other units. For the 10 kg hanging load, 98.07 N is 22.05 lbf and 10 kgf. The kg to Newtons Calculator converts a hanging mass straight into the tension it produces.

Tension is a contact force, and it is called a pulling or tensile force. Inside the rope it acts along the length, and the rope pulls on both ends with the same size of force. The Newton’s Third Law Calculator covers equal and opposite pairs like that one.

To measure tension, put a spring scale, a load cell or a dynamometer in line with the rope and read the force directly. A spring scale works by stretch, which the Hooke’s Law Calculator explains.

A tension meter, or tensiometer, measures a line that cannot be cut or unhooked. It clamps onto the cable and bends it slightly between three points. The force needed to make that bend tells the meter the tension in the line.

String, Wire and Bolt Tension

Several other things go by the name tension. The types that come up most are static tension in a hanging rope, tension in an accelerating rope and tension at an angle. Outside rope problems there are also string tension on instruments, bolt tension and surface tension in liquids.

String Tension on a Guitar or Racket

A guitar string tension calculator works from pitch, not from a hanging load. D’Addario gives the tension in pounds from the string’s unit weight UW in lb/in, the scale length L in inches and the frequency F in hertz.

$$T = \frac{UW \, (2 L F)^{2}}{386.4}$$

Tennis string tension is a set pull applied by the stringing machine, quoted in pounds or kilograms. A racket strung at 55 lb has 244.7 N in each string, or 24.95 kgf. Neither case is a rope setup, so they use these formulas and not the tool.

How to Calculate Tension in a Wire

The tension in a wire is found the same way as in a rope, from the load and the angles. What differs is the limit. The maximum tension a wire can take is its tensile strength multiplied by its cross-section area.

The word also has an electrical meaning. High-tension lines are high-voltage lines, and tension there means voltage. The mechanical tension in those same conductors follows the sag rule given earlier for a horizontal cable.

How to Calculate Tension in a Bolt

Bolt tension, or clamp load, is estimated from the tightening torque. Portland Bolt uses T = K × D × P, where T is torque, D is the bolt diameter and P is the clamp load. K is a torque coefficient of about 0.20 for plain bolts and 0.25 for hot-dip galvanized ones.

Rearranged, P = T ÷ (K × D). A plain 1/2 inch bolt tightened to 50 ft-lb, which is 600 in-lb, has roughly 600 ÷ (0.20 × 0.5) = 6,000 lb of tension. That is about 26.7 kN.

How to Calculate Work Done by Tension

Work done by tension is the tension times the distance moved times the cosine of the angle between them. Pulling with 24 N at 60° over 3 m does 24 × 3 × cos(60°) = 36 J. A string that swings a ball in a circle does no work, because the tension is always at right angles to the motion.

For forces that are not tensions, such as a push, a pressure load or a spring force, the Force Calculator is the place to start. Every related tool is listed under Dynamics and Statics Calculators.