# Tension (physics)

In physics, tension is the force transmitted through a string, rope, cable, or any other type of flexible connector when it is pulled tight by forces acting from opposite ends. This force is directed along the length of the connector and pulls equally on the objects at either end. For example, when you hang a picture on a wall using a wire, the tension in the wire supports the picture’s weight and keeps it in place. Similarly, when two people play tug-of-war, the rope experiences tension as both sides pull on it. Tension is the opposite of compression, which is the force that pushes or squeezes material together. While tension pulls objects apart, compression pushes them together. Tension plays a vital role not only in engineering structures but also in everyday items like swings and elevators, ensuring their stability and functionality.

Contents

## Formula

The tension force formula, used to calculate the force exerted by a string, rope, or cable, is given by FT = mg ± ma, where FT represents the tension force, m is the mass of the object, g is the acceleration due to gravity, and a is the acceleration of the object. This formula allows for the accurate determination of the tension force acting on an object in various scenarios.

When an object is moving upward, the tension force is given by FT = mg + ma, taking into account the additional force required to overcome gravity and accelerate the object upward. Conversely, when the object is moving downward, the tension force is expressed as FT = mg – ma, accounting for the opposing forces of gravity and the acceleration of the object in the downward direction. Finally, when the object is at rest, the tension force is equal to the weight of the object, resulting in FT = mg.

### Practice problems

#### Problem #1

A wooden block with a mass of 2 kg is hanging from a rope. If the block is assumed to be accelerating upward at a rate of 25 m/s2, calculate the tension force acting on the block. Take the value of gravitational acceleration, g, to be 9.81 m/s2.

Solution

Given data:

• Mass of a wooden block, m = 2 kg
• Acceleration of a wooden block, a = 25 m/s2
• Tension force acting on a wooden block, FT = ?
• Gravitational acceleration, g = 9.81 m/s2

Applying the formula, when an object is moving upward:

• FT = mg + ma
• FT = (2 × 9.81) + (2 × 25)
• FT = 19.62 + 50
• FT = 69.62 N

Therefore, the tension force acting on a wooden block is 69.62 N.

#### Problem #2

An 8 kg block is attached to a string and is accelerating downward at a rate of 5 m/s2. Determine the tension force acting on the block.

Solution

Given data:

• Mass of a block, m = 8 kg
• Acceleration of a block, a = 5 m/s2
• Tension force acting on a block, FT = ?

Applying the formula, when an object is moving downward:

• FT = mg – ma
• FT = (8 × 9.81) – (8 × 5)
• FT = 78.48 – 40
• FT = 38.48 N

Therefore, the tension force acting on a block is 38.48 N.

#### Problem #3

What is the tension force acting on a rubber tire with a mass of 2 kg that is hanging from a rope? Assuming the rubber tire is stationary and not accelerating.

Solution

Given data:

• Mass of a rubber tire, m = 2 kg
• Tension force acting on a rubber tire, FT = ?

Applying the formula, when an object is at rest:

• FT = mg
• FT = 2 × 9.81
• FT = 19.62 N

Therefore, the tension force acting on a rubber tire is 19.62 N.

#### Problem #4

Calculate the tension force acting on a 4 kg box in the following scenarios:
(a) When the box is moving upward with an acceleration of 8 m/s2.
(b) When the box is moving downward with an acceleration of 6 m/s2.
(c) When the box is at rest.

Solution

Given data:

• Mass of a box, m = 4 kg
• Tension force acting on a box, FT = ?

(a) When the box is moving upward with an acceleration of 8 m/s2

• Acceleration of a box, a = 8 m/s2

Applying the formula, when an object is moving upward:

• FT = mg + ma
• FT = (4 × 9.81) + (4 × 8)
• FT = 39.24 + 32
• FT = 71.24 N

Therefore, the tension force acting on a box is 71.24 N.

(b) When the box is moving downward with an acceleration of 6 m/s2

• Acceleration of a box, a = 6 m/s2

Applying the formula, when an object is moving downward:

• FT = mg – ma
• FT = (4 × 9.81) – (4 × 6)
• FT = 39.24 – 24
• FT = 15.24 N

Therefore, the tension force acting on a box is 15.24 N.

(c) When the box is at rest

Applying the formula, when an object is at rest:

• FT = mg
• FT = 4 × 9.81
• FT = 39.24 N

Therefore, the tension force acting on a box is 39.24 N.