We have all been to parks and recreational centres. Often there are a lot of swings for us to enjoy. We can sit on the seat, and swing ourselves back and forth and enjoy a smooth breeze. The swings are operational due to the Tension Force. The tension formula can be represented as \(T = mg + ma\).

Tension occurs due to the direction of pull on the object and along the length of the given rope. Furthermore, it is important to note that Newton is the SI unit of Tension. In this article, we will discuss about tension formula in detail. Scroll down to find more about Tension and its formula!

What is Tension Force?

In Latin, the word “tension” means “to stretch”. The tension force is the force that acts along the length of a flexible medium like a rope, or cable, or chain. We know that a force can be a push or a pull. In physics, we deal with several kinds of forces like weight, normal force, thrust, friction etc. A force based on how it acts and transmits can be either a contact or a non-contact force. Tension force is a contact force. It is carried along the length of a flexible medium.

Commonly known as “action-reaction pair” force, tension acts at each element of the flexible medium. If we consider any cross-section of a rope, part of the rope on one side to the cross-section will apply action force on the part of the rope on the other side to the cross-section. Similarly, the second part of the rope exerts reaction force on the first part. Therefore at any cross-section, we can see tension force acting in both directions. At endpoints, the rope will apply tension force on the object connected to it towards itself (pulling force), and the object will apply reaction force on the rope towards itself. Directions of these forces are along the length of the rope.

Facts about Tension Force

1. Tension: It is often seen that we cant push effectively using ropes, but when it comes to pulling on objects, ropes or such cables are very effective. This is because tension is a pull force and not a push force. As we try to push an object using a rope, it will get slack and thus lose its tension. Tension only acts when the rope is taut.  This is a common mistake that people make while drawing the tension force in an FBD. The tension force can not push against a body; it can only be used to pull it.
2. Tendons: We have muscles named “tendons” in our bodies. These muscles are flexible and carry forces to other parts of the body, parallel to the lateral length of the muscles. The tension transferred between these tendons carries the force along the length of our body parts.
3. We know that when we pull the molecules apart at the molecular level in an object, these molecules gain potential energy. Due to this, a restoring force is developed within the molecules, and this restoring force generates tension. In the case of flexible mediums, like ropes or cables or strings, this tension force tries to pull the molecules back to their original position and restore the medium to its original length. We generally represent the tension force by \(T\) or \(F_T\). 
4. The force due to tension exerted by a massless rope connecting two objects will be the same on both objects.
5. Tension acts along the direction of pull on the object and along the length of the given rope.
6. Since tension acts in both directions on all elements at any given point in a flexible medium, the work done by tension is positive on one side and negative on the other side. Thus, net work done by tension on any element of the flexible medium is always zero.

Tension Formula

Let’s calculate the result of tension force, for an object hanging from a massless rope as shown in the figure below:

The object’s weight will act downwards while the tension force of the rope will try to pull the mass up. Let \(m\) be the mass, \(W\) be the weight of the object, \(g\) be the acceleration due to gravity, \(a\) be the acceleration of the object and \(T\) be the tension force, then, based on the state of motion of the object, three cases are possible:
a. When the object is at rest: Tension force will balance the weight of the object, thus:
\(T = W\)
\(T = mg\)
b. When the object is moving downwards with acceleration: From Newton second law of motion, the net force acting on the object is equal to its mass multiplied by its acceleration. Since the object is accelerating downwards, weight must be greater than tension. Therefore,
\(W – T = ma\)
\(mg – T = ma\)
\(∴ T = mg – ma\)
c. When the object is moving upwards with acceleration: Since the object is accelerating upwards, weight must be lesser than tension. Therefore,
\(T – W = ma\)
\(T – mg = ma\)
\(∴ T = mg + ma\)
As we can see, the result for the tension is case dependant. We can follow methods similar to those shown above, to get results for tension.
The SI unit of tension is newton \((\rm{N})\).
The dimensions of tension are \([M^1 L^1 T^{-2}]\).

How to Calculate Tension Force?

From our understanding of the tension force, it can be stated that there is no specific formula to calculate the tension force exerted by a rope or cable. We calculate tension force in the same way we calculate the normal force, i.e. by newton’s second law of motion. Identify all the forces acting on the given body and use them to calculate tension. To do so, follow these steps:
Step 1: Draw the body diagram of the body on which forces are acting.
Step 2: Select a direction, and apply the equation of newton’s second law of motion \(\left( {a = \sum {\frac{F}{m}} } \right)\) accordingly.
Step 3: Solve this equation to calculate the tension force.

Solved Examples on Tension Formula

Q.1. A \(10\,\rm{kg}\) mass is dangling at the end of a thread. If the acceleration of the mass is acting as:
(a) \(5\,\rm{m/s}^2\) in the upward direction.
(b) \(5\,\rm{m/s}^2\) in the downward direction.
Calculate the tension in the thread.
Ans: We are given:
Mass of the hanging body, \(m = 10\,\rm{kg}\)
\(g = 9.8\,\rm{m/s}^2\)
\(a = 5\,\rm{m/s}^2\)
(a) When the body is accelerating upwards, the tension force is
\(T = mg + ma\)
\(T = 10(9.8 + 5)\)
\(T = 148\,\rm{N}\)
(b) When the body is travelling downwards, the tension force is:
\(T = mg – ma\)
\(T = 10(9.8 – 5)\)
\(T = 48\,\rm{N}\)

Q.2. A monkey climbs up a cable tied to a tree with an acceleration \(2\,\rm{m/s}^2\). If the mass of the monkey is \(10\,\rm{kg}\). Find the tension in the cable. Take \(g=10\,\rm{m/s}^2\).
Ans: We are given,
Mass of monkey: \(m = 10\,\rm{kg}\)
Acceleration of monkey: \(a = 2\,\rm{m/s}^2\)
The tension in the cable will be equal to the apparent weight of the monkey, thus:
\(T = mg + ma\)
\(T = 10(10 + 2)\)
\(T = 120\,\rm{N}\)

Summary

The tension force is the force that acts along the length of a flexible medium like a rope, or cable, or chain. Tension force is a contact force. It is carried along the length of a flexible medium. Commonly known as “action-reaction pair” force, tension acts at each element of the flexible medium. If we consider any cross-section of a rope, part of the rope on one side to the cross-section will apply action force on the part of the rope on the other side to the cross-section.

Tension formula can be represented in several ways depending on various conditions. If the object is at rest then it is represented as \(T = mg\). If the object is moving upwards with acceleration then it is represented as \(T = mg + ma\). If the object is moving downwards with acceleration then it is represented as \(T = mg – ma\). The SI unit of Tension is Newton.

FAQs on Tension Formula

Q.1. What is the Tension force?
Ans: The pulling force exerted axially by a rope or cable along its length is known as Tension force.

Q.2. Is tension a contact or a non-contact force?
Ans: Tension acts in a rope when the surface is in direct contact with the rope. Thus, it is a contact force.

Q.3. Does gravity affect the tension force?
Ans: Tension force is due to the electromagnetic forces developed in the rope when it is taut. It is independent of gravity. The rope under tension pulls the objects connected it its two ends. The magnitude of the tension depends on the force with which the objects connected to it are pulling. Therefore, if the force exerted by the objects is due to gravity, tension force indirectly depends on gravity. In other situations, it is independent of gravity.

Q.4. What is the SI unit of tension?
Ans: The SI unit of Tension is Newton.

Q.5. Suppose two blocks of mass \(2\,\rm{kg}\) each are suspended by a string. What will be the tension force in the string? Take \(g = 10\,\rm{m/s}^2\).
Ans: Tension will balance the weight of the two masses.
The weight of the two blocks, \(W = mg + mg = 2mg = 40\,\rm{N}\)
Thus, the tension in the rope, \(T = W = 40\,\rm{N}\).

Q.6. What is the formula for tension?
Ans: The formula for tension is \(T = mg + ma\).

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