Dimensional Formula of Tension

As per Newton’s laws of motion, force is an external force that can change the body’s state in rest and activity. However, this particular explanation doesn’t define the entire phenomenon. Therefore, force is an external agent that can cause energy buildup or displacement in a body. There are two major types of forces: contact and non-contact forces. From the name itself, you can understand that the non-contact forces exist between two bodies that aren’t in physical contact, like the gravitational pull between heavenly bodies. On the contrary, contact forces are defined as forces existing between two material bodies in touch, as the tension force. 

What is the Tension Force, and How is it Developed?

Tension force is a type of contact force that is usually generated in the ropes, strings, and other such elements. When you keep a rope in an open place, with both the ends kept free, no tension force is developed. However, when the rope’s one end is attached to a bar, the other is attached to a load, and the entire arrangement is hung, the rope’s tension force develops. 

Several explanations can define the formation of this force element. 

1. When a load is hung from the rope, the main force acting on it will be its weight, W, and it will work in the downward direction. 
2. Since Newton’s third law states that every action has an equal and opposite reaction, we can consider that a counterforce will develop in the rope to balance the arrangement. 
3. This force is termed tension, and it acts in the opposite direction of the weight, i.e., towards the upward direction. 

The Physical Expression of Tension Force

Tension forces are constantly developed whenever a rope, string, or anything like this is stretched. The stretching can be done in a vertical position or around an angle. Usually, the tension developed in the string is a pull force, and it always acts as an opposite component of a push force. Therefore, neither of the entities will have sole existence with the other one. 

Now, at any given point, both the push force and the tension need to be equal to ensure that the entire system is balanced and there is no gain or loss in acceleration. This is why no particular equation can explain the physical expression of tension force. It entirely depends on the situation you are considering. 

Hanging body from the rope

When a body is hung in the downward position from a rope, the pushing force acting on it is the weight expressed as W. Therefore, following the concept described above, and considering that there is no change in acceleration, the tension force generated will be equal to the weight. 

T = W

A stretched rope with an angle of 30˚

Consider a stretched rope attached to a body kept on a flat surface. The cord is elevated and attached to a point at an angle of 30˚ from the horizontal plane. In this case, the body’s weight is W acting downwards, and in response to it, the surface will exert an upward force, known as regular or N. 

Therefore, the horizontal component of the tension force, T, will be Tcos30˚, whereas its vertical component is Tsin30˚. To know the value of tension force, we need to consider the horizontal component. So, the physical expression will be: 

T cos30˚  = mass × acceleration

Or, T = (mass × acceleration) / cos30˚

Dimensional Formula of Tension Force 

As described above, no such concept can define a proper physical expression for the tension. It ultimately depends on the free-body or rigid body arrangement we are considering. However, there is a common expression that we can consider. 

Tension is nothing but a force, and hence, it will depend on mass and acceleration as per Newton’s second law. As it doesn’t always act in the straight vertical or horizontal direction, the cosine or sine component can get added if the force has an angle with any planes. However, these components do not have any dimension. 

So, from this, the following facts can be deduced: 

T = acceleration × mass

Or, T = m × a

Or, T = m × v/t

Or, T = [m × d] / t2

Or, T = [M1L1T-2] 

Conclusion 

Tension force is considered throughout since it is developed whenever a string, rope, and other is stretched. From the above notes, you can understand that tension depends on mass, length, and time. For example, if the body’s mass connected with one end of the rope is more, the tension developed in the string will also be more. Similarly, if we consider the rope’s length, the tension force will vary directly but as a square. Therefore, the tension forces play a crucial role in determining the resultant action of any arrangement.