How to find tension in a string? | Tension formula in physics

If you drop a ball from the roof of your house, the ball will fall under gravitational force. Now, attach the ball to an end of a string and fix the other end of the string somewhere on the roof. Then drop the ball from the roof again. This time the ball will be hanging below the roof. We can understand that the gravitational force is still acting on the ball in the downward direction. But, how is the ball balanced while hanging? Is there any force in the opposite direction of gravity? The answer is ‘yes’ and this opposite force is the tension of the string. In this article, we are going to discuss the concepts of tension in the string, the tension formula, and how to find tension in a string.

Contents of this article:

1. What is the tension of a string?
2. Unit of tension
3. Formula of tension of a rope
4. How to find string tension?
5. Numerical problems on tension of string

What is the tension in a string?

Tension is a type of force (pulling force) that appears along the length of a string or rope when an external force acts at one of the ends of the rope. The tension of string acts in the opposite direction of the external force.

Units of tension

The tension of a rope is one kind of force. So, the units of tension are the same as those of force.

Therefore, the SI unit of tension is Newton (N) and the CGS unit is dyne.

Formula of Tension in a string

As I already have mentioned that the tension of a string appears when an external force acts at one of its ends. Now, the external force can arrive in various forms, like gravitational force, centripetal force, horizontal pulling force, etc. But this is sure that the tension force will act opposite to every applied force and at the balanced condition, the tension force of the string will be equal to the externally applied force. So, we will have different formulae for tension in a string depending upon the different types of external forces.

1. When the external force is a horizontal pulling force (F). Then tension of the string at the balanced condition is T = F.
2. If we rotate a particle in a circular path after attaching it to a rope, the tension force of the rope will balance the centripetal force. If a particle of mass m is rotating along a circular loop of radius r with a speed v, then the centripetal force on the particle is, \small {\color{Blue} F=\frac{mv^{2}}{r}}. Then the formula for tension of the string or rope is \small {\color{Blue} T=F=\frac{mv^{2}}{r}}.
3. When the string helps to hang an object falling under gravity, then the tension force will be equal to the gravitational force. If an object of mass m is falling under gravity, then the tension of string will be equal to the weight of the object i.e. tension, T = mg, where, g is the acceleration due to gravity.

These are the basic idea of the tension formula in physics.

How to find tension in a string?

Now, it’s time to know how to calculate the tension in a string. There can be two situations to calculate the tension force of a string or rope – one is at the equilibrium condition under multiple forces and another one is in an accelerated rope-mass system.

1. One can easily find the tension of a rope at mechanical equilibrium by using the above three formulae of tension force.
2. But, for an accelerated system, we need to know the mass and acceleration of the block or body attached and other forces on the block (if any). Then we need to develop an equation for the resultant force on the block by using Newton’s force equation of motion. Then we will be able to find the tension of the rope on the block from this equation. Check the numerical problems below for a better understanding.

Numerical problems on string-mass system

Question-1: A body of mass 15 kg is hanging at the lower end of a lightweight and inextensible string. If the other end of the string is tightly attached to a hook, then what is the tension of the string acting on the body?

Answer: The mass of the body, m = 15 kg. The body is not falling. So, it does not have any acceleration. Thus, the tension of the string balances the weight of the body. Hence, tension of string, T = mg = 15×9.8 = 142.5 Newton.

Question-2: A monkey of mass 12 kg climbs up a vertical rope that is inextensible and has negligible mass. If the acceleration of the monkey is 2 m/s² in the upward direction, then find the tension in the string. In which direction the tension will act? [take g = 10 m/s²].

Answer: The mass of the monkey, m = 12 kg, acceleration, a = 2 m/s². Here, the tension force and the gravitational force are not balanced. Because there is an acceleration that is caused by the resultant upward force. This force is coming from the effort of the monkey to climb up. Since the gravitational force is acting downwards, then the tension in the rope will arise in the upward direction.

Then from Newton’s second law of motion, the resultant force on the monkey is,

F = (T – F_g) [F_g is gravitational force]

or, T = (F + F_g)

or, T = (ma + mg) [Gravitational force, F_g=mg]

or, T = m (a +g)

or, T = 12×(2 + 10)

or, Tension in the rope is T = 144 Newton.

This tension force will act in the opposite direction of gravitational force i.e. tension is along the upward direction.

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This is all from this article on the tension of a string and tension formula in physics. Hope you understand how to find the tension in a string in different situations. If you have any doubts on this topic you can ask me in the comment section.

Thank you!

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1. Work done by tension force
2. Equilibrium of forces
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4. Block-Pulley systems
5. Newton’s second law of motion