The Force of Tension

In this science blog we investigate the force of ‘tension’. We look at tension along a rope, tension in the chain of a hanging light and tension along an elevator cable.

Tension and horizontal pulling forces

Whenever a force is directed along a ‘linear object’ such as a rope, cable, wire or string that force is typically referred to as ‘tension’.

‘Tension force’ is transmitted when the ‘linear object’ is stretched by other forces which act at opposite ends and which pull in opposite directions.

Any tension force generated is evenly distributed along the length of the linear object.

two stick men pulling on rope showing extent of tension force

Tension forces are not produced in isolation from other forces as can be seen below:

Diagram showing all forces which act on two people pulling a rope

Example 1-Opposing pulling forces are balanced

Two teams take part in a tug of war contest. By pulling on the rope from opposite ends and in opposite directions, each team applies force to the rope. The greater the opposing pulling forces, the greater the ‘magnitude’ (or ‘quantity’) of tension force along the rope.

tug of war teams each with four contestants showing tension of greater magnitude

The smaller the opposing pulling forces, the lesser the ‘magnitude’ of the tension force per unit area of the rope.

tug o'war with two people pulling a rope with tension force of a lesser magnitude

The ‘magnitude’ of any tension force is measured in Newtons.

  • What is the tension force along the rope in this example? To answer this question we need to refer to Newton’s First and Third Laws of Motion.

tension force of rope where two teams pull; with force of 250N

Newton’s First Law of Motion

An object at rest stays at rest and an object in motion stays in motion with the same speed and direction unless acted on by an unbalanced force.

The blue team remains ‘at rest’ because the pulling forces acting on it are balanced. Likewise the red team remains ‘at rest’ because the pulling forces acting on it are also balanced.

Newton’s Third Law of Motion

two pears on a dish GW Platt

Forces come in ‘pears’

For every action force there is an equal and opposite reaction force.

Not only are the forces acting on the red and blue teams balanced but the forces acting on the rope are also balanced.

If equal and opposite pulling forces acting on both the red and blue teams is 250N, then the tension force acting on the rope between the two teams must also be 250N.

Forces always come in ‘pears’- equal and opposite action-reaction ‘pairs’.

Tension force along rope where pulling forces are in equilibrium

 Example 2- opposing pulling forces are unbalanced

In the next example two additional people join the blue team.

  • What will the tension force be? We now need to consider Newton’s Second and Third Laws of Motion.

Net force of 100 n towards winning team in tug o war contest and asked to consider the tension force along the rope

Newton’s Second Law of Motion

The acceleration of an object is dependent on the net force acting upon the object…..

spring-scale balance

Spring Scale

At the same time as the blue team pulls with a force of 350 N, the red team pulls with a force of 250N. The greater force acting on the red team means that the red team accelerates towards the left and loses the tug o’war contest. There is a net force of 100N acting on the red team.

Newton’s second law refers to what makes a single object accelerate. It is the unbalanced forces acting on the red team which makes the team accelerate.

The Second Law informs us about the motion of the tug of war teams and not the tension force along the rope.

Newton’s Third Law of Motion

For every action force there is an equal and opposite reaction force.

The tension force between the two teams remains unchanged. An action force of 250N is still met by an equal and opposite reaction force of 250N.

The extra force with which the the blue team pulls the rope is translated into the leftwards acceleration of the red team, not increased magnitude of tension in the rope.

tension force revealed when two unbalanced forces pull a rope in tug of wat contest

To prove that the tension force remains unchanged when the opposing ‘pulling’ forces are unbalanced a spring scale can be attached to the center of the rope.

The spring scale would read the same even when it is turned round to face the opposite direction.

spring balance included in the rope and switched around to show how the reading is the same

Take another example of unbalanced pulling forces; a strongman and a small girl pull a rope from opposite ends in opposite directions. The girl pulls with a force of 50N while the strongman pulls with a force of 150 N.

strongman girl pulls girl with force of 150N girl pulls strongman with force of 50N in tug of war

The strongman pulls the rope with a force greater than the girl; the girl accelerates towards the left because there is a force acting on her. (Newton’s Second Law)

In this situation the tension force along the length of the rope between the two objects (the strongman and the girl) will be 50N. (Newton’s Third Law- there is an action-reaction force pair.)

The mismatch of pulling forces results in the accleration of the girl towards the left and not extra tension in the rope.

Example 3 -friction and tension force along a rope

In the next example the strongman stands on ice while the girl stands on solid ground.

The strongman pulls the rope with a force of 150 N. However the absence of friction on the ice means that the strongman now accelerates towards the right. There is a net force now acting on the strongman.

Strongman pulls rope on ice and net force towards girl

The tension along the rope remains the same at 50N.  The force between the strongman and the girl remains balanced.

Example 4-Pulling a rope tied to a tree

In this next example a man pulls a rope tied to a tree. Just as the man pulls the tree with a force of 15oN, the tree also ‘pulls’ the man with an equal and opposite force of 150N. (Newton’s Third Law of Motion – for every action there is an equal and opposite reaction)

  • What is the magnitude of the tension force?

Rope tied to a tree and man pulling tree with a force of 150N

The magnitude of the tension force will be 150N.

  • What will the tension force be if the rope now breaks?

man pulls rope which snaps and no longer any tension force in rope
The forces are now unbalanced since the tree no longer ‘pulls’ the man. The man accelerates backwards and falls to the ground. There is no longer any tension in the rope.

Tension and ‘vector quantities’

A car traveling west at 20 mph has both speed and direction. The motion of this object, described by a numerical value of 20mph and a westerly direction is a ‘vector quantity’.

three wheeler car travelling west at 20 mph explains a vector quantity

All forces are ‘vector quantities’ and tension is no exception.

  • Where is the direction of travel of a rope under tension?

There is no direction of travel but that does not stop the force of tension in the rope being a vector quantity. The equal and opposite forces cancel each other out producing a sum of forces between the two ends of the rope of zero newtons.

sum of tension force is zero so tension ios a vector force

Vector quantities can be contrasted to scalar quantities which just give an indication of ‘magnitude’ with no indication of any direction of travel.

Metric_speedometer_from_a_1992_Euro-spec_Passat_B3

2) Tension and gravity

Example 1- equal mass 

Gravity also affects the magnitude of any tension force.

This spring balance measures the magnitude of the tension exerted on a wire by the gravitational force of two equal weights pulling at opposite ends in opposite directions.

  • What will the reading on the spring balance be?

spring scale between two weights measures tension

The spring balance will read 100 N since the action and reaction forces are 100N. Both weights remain at rest; the net force (or ‘sum of forces’) is zero.

Example 2- unequal mass

In the next example one of the 100N weights is substituted for a 200N weight. The net force acting on the remaining 100N weight is now 100N.

Unbalanced forces as 200N weight pulls 100N weight by force of gravity. tension force remains at 100N

The spring scale does not measure the downwards acceleration. It simply measures the tension, the magnitude of the force exerted on the wire. The tension force of the spring balance will still only read 100N.

Example 3- light hanging from ceiling

In this example the downwards force exerted on the chain by the weight of the lamp is 50N. The upwards ‘normal’ force exerted on the chain by the ceiling is also 50N.

We can say that the upwards force must equal the downwards force.The tension along the chain equals the weight. (Newton’s Third Law)

Upwards force of ceiling in equilibrium with downwards force of light showing magnitude of tension in chainux1.eiu.edu/~cfadd/1150/04Nwtn/appl.html

Example 4 -tension in cable of an elevator

We now investigate what happens to tension in a vertical cable when the mass it is supporting- in this instance an 800 kg elevator, is at rest and in motion.

When this 800kg elevator is at rest the upwards and downwards forces are in equilibrium. (Newton’s First Law)  The tension in the cable is 7840N.

800 kg elevator at rest with gravity and pulling force equal the tension force is 7840 N

We know that the tension of the cable holding an elevator weighing 800 kg because the math tells us so!

Calculating the tension of a cable holding an elevator with a mass of 800 kg plaza.obu.edu/corneliusk/up1/fec.pdf

When the 800 kg elevator either ascends or descends at a constant velocity, with the upwards and downward forces again in equilibrium, the tension force along the cable will still be 7840N.

elevator ascending or descending at a constant velocity tension force in cable remains 7840 N

The tension in the cable decreases to 6240N when the elevator accelerates downwards from rest. The downwards force of gravity is greater than the upwards pulling force. Less work needs to be done to counter the affects of gravity.

Tension in cable is 6240 N when elevator accelerates downwards

The tension in the cable increases to 9440N when the elevator accelerates upwards from rest. The force pulling the elevator upwards is greater than the downwards force of gravity. Greater work must be done to overcome the force of gravity

elevator accelerates upwards from rest and tension in cable is 9440N

How does a cable or rope gain tension?

Tension is produced at the atomic level. When a cable or rope is pulled apart the chemical bonds holding the atoms together  s-t-r-e-t-c-h apart.

tension in a solid as interatomic chemical bonds are stretched

Links

Experiment with tension forces by playing this interactive science game.

Notice how a player in this tug of war contest gains a competitive advantage by stepping onto the grass and increasing the backwards force of friction.

Science fun stuff-A real tug of war contest!

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