Question

What is the formula of tension?

Answer 1

Hint: Tension is a force that has a pulling back tendency. It acts along the complete length of string or cable. Under different conditions, tension can be different. In this question, we shall analyze the tension in every case possibly existing and then equate the forces acting in opposite directions to maintain the equilibrium.

Complete step by step answer:
Tension is a force that acts along a uniform continuous length of the cable or string. It is usually transmitted axially and is described as one of the forces in the action reaction pair being formed in a particular situation.
 Let’s consider the following situations
Here W is the weight of the body and m is the mass of the body
For equilibrium to exist, we shall equate the forces acting in the opposite directions.
Case (i) If the body is moving upwards with an acceleration a and the tension also acts upwards then we can say that $T = W + ma$
Case (ii) If the body is moving downwards with an acceleration a and the tension acts upwards then we can say that $T = W - ma$
Case (iii) If the body is just suspended (not moving) the tension will be given as $T = W$
Case (iv) If the body moves up or down with uniform speed, it means that the tension can be given as $T = W$
 This can be generalised as the mathematical expression given below where the tension depends on the acceleration of the moving object.
$T = W \pm ma$
 The weight of the object is $W = mg$ where m is the mass of the body and g is the acceleration due to gravity.
 Therefore, the formula for tension can be modified as:
$T = m(g \pm a)$
 Where, m is the mass of the body, g is the acceleration due to gravity, a is the acceleration of the moving body.

Note: The tension acting in a string generally remains constant throughout the length of the string. However, in cases where the mass density of the rope or the string varies, the tension is different at every point. So, we need to calculate the tension by writing force equations for every point and balancing them to maintain equilibrium. When anything is not mentioned in the question about the mass density, we assume it to be constant.