Tension Calculator

Tension Scenario:

Number of Ropes:

Gravitational:

kg ▾

Gravitational:

m/s² ▾

Angle α:

deg ▾

Angle β:

deg ▾

Tension Calculator:

This tension calculator finds the tension force in a rope, cable, chain, or string when lifting, pulling, or supporting an object. It determines the amount of force acting in the system under different conditions, such as hanging masses, angled ropes, pulleys, or objects moving on a surface. With this tool, you can make tension calculations in physics for:

1 or 2 strings used to lift the object against gravity
String used to pull 1, 2, or 3 objects on a surface

What is Tension in Physics?

Tension is a stretching or pulling force that is transmitted axially along an object, such as a rope, cable, chain, etc., to pull an object. It is a special kind of force that acts in a direction opposite to the compression and acts on opposite ends of the rope.

How to find Tension (Force) in Ropes Suspending an Object?

Tension in a cable (due to an object hanging) can be classified in two cases:

Single Rope

Suppose an object is lifted with a string, as shown in the above picture. In this condition, the tension in the string is equal to the weight of the object due to gravity (approximately 9.8 m/s²). Now, if we consider the upward force as positive and the downward force as negative, they cancel each other’s effect, and the overall sum of both forces equates to zero.

ΣF↑ = 0 = T + (-W) T = W

where;

W is the weight acting downward
T is the upward tension force

Tension in Multiple Ropes

This is a bit more complex case, where the tension is distributed along two ropes used to suspend an object of hanging mass 'm'. This force has influence along the horizontal and vertical components of the force.

T_1x and T_2x are the horizontal components
T_1y and T_2y are the vertical components

As the gravitational force acts vertically downwards, we will only consider the vertical components of the pulley tension force, such that:

Σ F↑ = 0; T_1y + T_2y + (-W) = 0

Moving '-W' to the other side of the equation:

W = T_1y + T_2y

The components of the angle along T_1y and T_2y can be expressed in terms of T₁ and T₂, such that:

T_1y = T₁ * sin(α)

T_2y = T₂ * sin(β)

W = T₁ * sin(α) + T₂ * sin(β) --- (1)

Now, regarding the horizontal components, there is no movement in this direction because the whole system is in static equilibrium. It shows that both x components are equal.

T_1x = T_2x or T₁ * cos(α) = T₂ * cos(β)

Moving cos(α) to the other side:

T₁ = (T₂ * cos(β)) / cos(α)

Putting the value of T₁ in equation (1):

W = T₁ * sin(α) + T₂ * sin(β)

W = T₂ * [(cos(β) / cos(α)) * sin(α)] + T₂ * sin(β)

W = T₂ * [(cos(β) * sin(α) / cos(α)) + sin(β)]

T₂ = W / [ (cos(β) * sin(α)) / cos(α) + sin(β) ]

Now we have;

T₁ = (W / [ (cos(β) * sin(α)) / cos(α) + sin(β) ]) * [ cos(β) / cos(α) ]

The given formula is considered to calculate tension (force), considering the tension formula with the angle of its orientation.

T₁ = W / [ (cos(α) * sin(β)) / cos(β) + sin(α) ]

Tension Cases for Dynamic Equilibrium:

In dynamic equilibrium, the value of acceleration (a) is not zero. In this condition, tension in a string has variable cases, including:

Motion of the Object	Rope Tension (T)
Moving Upward with Acceleration (a)	T = W + ma
Moving Downward with Acceleration (a)	T = W - ma
Suspended (Not Moving)	T = W
Moving Upward or Downward at Uniform Speed	T = W

Our tension calculator also considers these cases to help you find tension in a string (cable) under a dynamic equilibrium state.

How to find Tension in Ropes While Pulling an Object?

Pulling an Object Image

Example #1:

A mass of 10kg is attached to a string and pulled against a frictionless surface at an angle of 35°.

What is the tension in the string?

Tension example

Solution:

Step 1: As there is no frictional force, so tension will be equal to the gravitational force, such that:

T = f_g

Step 2: Write the expression for tension in the string.

T = m * g * sin(θ)

Step 3: Input given values to solve for the result.

T = 10 * 9.8 * sin(35°)

T = 56.154 N

Example #2:

Tension Example 2

Problem: Tension in a String for a Hanging Tire

Find the tension in a string that is used to hang a tyre of 30kg at a height of 15m?

Solution:

Step 1: Find the total of all forces.

F_g = m * g

F = T + F_g

F = T + (30kg) * 9.8ms^-2

F = T - 294N

Step 2: Identify acceleration. Since the tyre is not moving, there is no acceleration.

Step 3: Determine the tension force:

F = T - 294N

30 * 0 = T - 294N

T = 294N

Instead of manual calculations, you can solve your examples by simply adding the values into the tension calculator.

FAQ’s:

Is tension a contact force?

Yes, it is. When you tie an object of a certain hanging mass with a string, rope, cable, or similar medium, an internal pulling force is generated in the string that helps connect the object and the reference point. This is why tension is regarded as a contact force.

Can tension be negative?

No, the tension force itself cannot be negative because it is a pulling force and is represented by a positive quantity. However, while working with physics problems, you may find tension with a negative sign, which shows the tension is acting in the opposite direction of the motion or coordinate axis. The negative sign indicates the direction, it does not mean a negative tension.

Why is the work done by tension always zero?

As we know, the work equation:

W=F*s cosθ

If the tension in a string does not cause any displacement, then ‘s=0’

θ = 90∘ ⇒ cos⁡90∘ = 0

Therefore:

W=F*s cos⁡90∘ = 0

Hence, the work done by tension in this case is always zero.

How does gravity affect the force of tension?

The tension and gravitational forces act in opposite directions. Now, if the hanging object is not balanced by tension, it will accelerate towards the ground due to the force of gravity.

Tension Calculator

Tension Calculator:

What is Tension in Physics?

How to find Tension (Force) in Ropes Suspending an Object?

Tension Cases for Dynamic Equilibrium:

How to find Tension in Ropes While Pulling an Object?

Example #1:

Solution:

Example #2:

Problem: Tension in a String for a Hanging Tire

Solution:

FAQ’s:

Is tension a contact force?

Can tension be negative?

Why is the work done by tension always zero?

How does gravity affect the force of tension?

References: