Tension Formula

Tension Formula

Tension is a fundamental concept in physics and engineering, describing the force transmitted through a string, rope, cable, or any other type of flexible connector. It arises whenever an object is pulled or stretched by forces acting in opposite directions along its length. Tension plays a crucial role in various physical phenomena and engineering applications. It determines the behavior of structures under load, the stability of suspended objects, and the motion of objects connected by flexible links. Learn more about Tension, its definition, formula and examples in this article provided by Extramarks

What is Tension?

A force called tension acts along the length of a medium, particularly when the medium in question is flexible, such as a rope or cable. Tendons are flexible cords that transfer muscle forces to other body parts. Any flexible connector, such as a string, rope, chain, wire, or cable, can only be pulled in a direction perpendicular to its length. Consequently, the force that is transmitted by a flexible connector is tension with respect to the connector.

Tension is typically measured in units of force, such as newtons (N) or pounds-force (lbf). The magnitude of tension depends on factors such as the strength and stiffness of the connector and the forces applied to it.

The Dimensional formula of Tension is [M¹L¹T⁻²]

Formula for Tension

Since tension is nothing more than the drawing force that is present when the body is suspended, Its formula will then be as follows:

T = W±ma

Where,

• W = The body’s weight
• a = The speed of the moving body
• m = The body’s mass

In the above formula,

• The tension will be T = W + ma if the body is moving upward.
• The tension will be T = W – ma if the body is moving downward.

As a result, T = W if the tension is equal to the body weight.

The tension force acting on any object can be determined using the tension formula. It helps with a variety of mechanical issues. Since tension is a force, the Newton is its unit (N).

Solved Examples using Tension Formula

Example 1: An 8 kg mass is dangling at the end of a thread. If the acceleration of the mass is acting as:

(a) 3 m s⁻² in the upward direction.

(b) 3 m s⁻² in the downward direction.

Then determine the tension in the thread.

Solution:

Known parameters are:

Mass of the hanging body, m = 8 Kg,

g = 9.8 m s⁻²

(a) Given as a = 3 m s⁻²

If the body is traveling in the upward direction, the tension force is:

T = mg + ma

= 8×9.8+8×1.5

= 90.4 N

Example 2: Consider a crane lifting a load with a mass of

$500 \, \text{kg}$

500kg using a steel cable. The crane exerts an upward force on the cable to lift the load. If the load is lifted at a constant velocity, determine the tension in the cable.

Given:

• Mass of the load,
  $m = 500 \, \text{kg}$ 

  $m=500\text{kg}$

• Acceleration due to gravity,
  $g = 9.8 \, \text{m/s}^2$ 

  g = 9.8m/s²

Solution:

At constant velocity, the net force acting on the load is zero. The tension in the cable balances the weight of the load:

$T-mg=0$

$T = mg = (500 \, \text{kg}) \times (9.8 \, \text{m/s}^2) = 4900 \, \text{N}$

T = mg = (500 kg)×(9.8 m/s²) = 4900N

The tension in the cable holding the suspended load is

$4900 \, \text{N}$

4900 N

Example 3: Suppose a box of mass $m=10\text{kg}$ is suspended by a rope. The box is at rest, and the tension in the rope is to be determined.

Given:

• Mass of the box,
  $m = 10 \, \text{kg}$ 

  $m=10\text{kg}$

• Acceleration due to gravity,
  $g = 9.8 \, \text{m/s}^2$ 

  g = 9.8 m/s²

Solution: Since the box is at rest, the forces in the vertical direction must balance:

 T − mg = 0

Where:

• 
  $T$ 

  $T$ = Tension in the rope

• $m$ = Mass of the box
• $g$ = Acceleration due to gravity

Solving for

$T$

T:

$T = mg = (10 \, \text{kg}) \times (9.8 \, \text{m/s}^2) = 98 \, \text{N}$

T = mg = (10 kg)×(9.8 m/s²) = 98 N

The tension in the rope holding the box is

$98 \, \text{N}$

98N.