What Is the Equation for Tension in a Rope?

Rebecca Dongallo/CC-BY-2.0

The equation for tension in a rope is weight plus the product of mass and acceleration.

What Is Tension?

Every physical object that’s in contact with another one exerts forces. Depending on the objects that are making contact, the contact force has a different name. For rope, as with cable and chain, the force is called tension. Any object that rope is used to pull, hang, swing or support is subject to tension. As ropes are not usually used to push an object, tension in this respect is a pulling force. This pulling force can be exerted over a certain distance, which equates to the length of the rope.

If you attach a rope to an object and pull it, the rope begins to stretch and eventually goes taut. This is because it is under tension. If you release the tension on the rope by stop pulling on it, it goes slack. If the tension becomes too great so that the rope cannot stretch any further, it will eventually break. In practical terms, this means you need to check the rope’s tension limit when attempting to lift or pull an object with it.

The formula for tension remains the same regardless of the body acting on the rope or the body that is being acted upon. For the equation of tension in a rope, weight (W) is equal to the mass of the object (m) multiplied by the acceleration of gravity (g). Because of this, the equation can also be given as T = mg + ma. As tension is a force, the results given through the formula are expressed in newtons and noted with the symbol N.

Tension on a Single Length of Rope

For a single length of rope, its tension is determined by the forces acting at either end of it. Any changes in the mass of the object the rope is attached to or the level of acceleration instantly changes the tension in the rope. There are various factors that affect the tension.

Gravity is a source of constant acceleration, even when the rope is still. In the previous formula of T = mg + ma, “g” is acceleration relating to the object’s gravity that the rope is attached to while “a” refers to any other type of acceleration relating to this object. When an object is being suspended by a rope, the object’s weight counts as a type of acceleration.

Rotational acceleration is another consideration. When an object is hanging from a rope and being swung around in a circular motion, centripetal force occurs. The faster the object rotates, the higher the centripetal force is exerted by the rope to keep the object moving.

Also, friction has to be taken into account. When a rope is used to pull an object along another surface, such as the ground, friction occurs between the surface and the object. This is transferred to the rope’s tension.

Tension on Multiple Lengths of Rope

A rope used in a pulley system is configured so two lengths of rope are created with just the one piece. Used to lift parallel loads, both lengths have the same amount of tension. When non-parallel loads are being lifted via a pulley system, the rope’s tension is changed as the gravity force on the weight has changed, as has the pulling force on the second length of the piece of rope.