NEWTON'S THIRD LAW: APPLICATIONS, EXPERIMENTS AND EXERCISES - PHYSICAL - 2025

The third law of Newton, also called action and reaction law states that when an object exerts force on another, the latter also exerts on the first a force of equal magnitude and direction and the opposite direction.

Isaac Newton made his three laws known in 1686 in his book Philosophiae Naturalis Principia Mathematica or Mathematical Principles of Natural Philosophy.

A space rocket receives the necessary propulsion thanks to the expelled gases. Source: Pixabay.

Explanation and formulas

The mathematical formulation of Newton's Third Law is very simple:

F ₁₂ = - F ₂₁

One of the forces is called action and the other is reaction. However, it is necessary to emphasize the importance of this detail: both act on different objects. They also do it simultaneously, although this terminology incorrectly suggests that action occurs before and reaction after.

Since forces are vectors, they are denoted in bold. This equation indicates that we have two objects: object 1 and object 2. The force F ₁₂ is the one exerted by object 1 on object 2. The force F ₂₁ is exerted by object 2 on object 1. And the sign (-) indicates that they are opposite.

By carefully observing Newton's third law, an important difference is observed with the first two: while they invoke a single object, the third law refers to two different objects.

And is that if you think carefully, interactions require pairs of objects.

That is why the action and reaction forces do not cancel each other out or are balanced, although they have the same magnitude and direction, but the opposite direction: they are applied to different bodies.

Applications

Ball-ground interaction

Here is a very everyday application of an interaction related to Newton's Third Law: a vertically falling ball and the Earth. The ball falls to the ground because the Earth exerts an attractive force, which is known as gravity. This force causes the ball to fall with a constant acceleration of 9.8 m / s ².

However, hardly anyone thinks about the fact that the ball also exerts an attractive force on the Earth. Of course the earth remains unchanged, because its mass is much greater than that of the ball and therefore experiences negligible acceleration.

Another notable point about Newton's third law is that contact between the two interacting objects is not necessary. It is evident from the example just cited: the ball has not yet made contact with the Earth, but it nevertheless exerts its attraction. And the ball on Earth too.

A force such as gravity, which acts indistinctly whether there is contact between objects or not, is called a "force of action at a distance". On the other hand, forces such as friction and normal require that the interacting objects be in contact, that is why they are called "contact forces".

Formulas taken from the example

Returning to the ball - Earth pair of objects, choosing the indices P for the ball and T for the earth and applying Newton's second law to each participant in this system, we obtain:

_Resulting F = m. to

The third law states that:

m _P a _P = - m _T a _T

a _P = 9.8 m / s ² directed vertically downwards. Since this movement occurs along the vertical direction, the vector notation (bold) can be dispensed with; and choosing the upward direction as positive and downward as negative, we have:

a _P = 9.8 m / s ²

m _T ≈ 6 x 10 ²⁴ Kg

Regardless of the mass of the ball, the acceleration of the Earth is zero. That is why it is observed that the ball falls towards the Earth and not the other way around.

Operation of a rocket

Rockets are a good example of applying Newton's third law. The rocket shown in the image at the beginning rises thanks to the propulsion of hot gases at high speed.

Many believe that this happens because these gases somehow "lean" on the atmosphere or the ground to support and propel the rocket. It does not work like that.

Just as the rocket exerts force on the gases and expels them backward, the gases exert a force on the rocket, which has the same modulus, but opposite direction. This force is what gives the rocket its upward acceleration.

If you don't have such a rocket on hand, there are other ways to check that Newton's Third Law works to provide propulsion. Water rockets can be built, in which the necessary thrust is provided by the water expelled by means of a gas under pressure.

It should be noted that the launch of a water rocket takes time and requires a lot of precautions.

Use of skates

A more affordable and immediate way to test the effect of Newton's Third Law is by putting on a pair of skates and propelling yourself against a wall.

Most of the time the ability to exert force is associated with objects that are in motion, but the truth is that immobile objects can also exert forces. The skater is propelled backwards thanks to the force that the immobile wall exerts on him.

The surfaces in contact exert (normal) contact forces with each other. When a book is resting on a horizontal table, it exerts a vertical force called normal on it. The book exerts on the table a vertical force of the same numerical value and opposite direction.

Experiment for children: the skaters

Children and adults can easily experience Newton's third law and verify that the action and reaction forces do not cancel out and are capable of providing movements.

Two skaters on ice or on a very smooth surface can propel each other and experience movements in the opposite direction, whether they have the same mass or not, thanks to the law of action and reaction.

Consider two skaters with quite different masses. They are in the middle of an ice rink with negligible friction and are initially at rest. At a given moment they push each other by applying constant force with the palms of their hands. How will they both move?

Two skaters propel each other in the middle of an ice rink. Source: Benjamin Crowell (Wikipedia user bcrowell)

It is important to note that since it is a frictionless surface, the only unbalanced forces are the forces that the skaters apply to each other. Although the weight and the normal act on both, these forces do balance, otherwise the skaters would accelerate in a vertical direction.

Formulas applied in this example

Newton's third law states that:

F ₁₂ = - F ₂₁

That is, the force exerted by skater 1 on 2 is equal in magnitude to that exerted by 2 on 1, with the same direction and opposite direction. Note that these forces are applied to different objects, in the same way that the forces were applied to the ball and the Earth in the previous conceptual example.

m ₁ to ₁ = -m ₂ to ₂

Since the forces are opposite, the accelerations they cause will also be opposite, but their magnitudes will be different, since each skater has a different mass. Let's look at the acceleration acquired by the first skater:

So the movement that happens next is the separation of both skaters in opposite directions. In principle the skaters were at rest in the middle of the track. Each exerts a force on the other that provides acceleration as long as the hands are in contact and the thrust lasts.

After that the skaters move away from each other with uniform rectilinear motion, as unbalanced forces no longer act. The speed of each skater will be different if their masses are also.

Exercise resolved

To solve problems in which Newton's laws must be applied, it is necessary to carefully draw the forces acting on the object. This drawing is called a "free-body diagram" or an "isolated-body diagram." The forces exerted by the body on other objects should not be shown in this diagram.

If there is more than one object involved in the problem, it is necessary to draw a free-body diagram for each of the objects, remembering that the action-reaction pairs act on different bodies.

a) The acceleration that each skater acquires thanks to the push.

b) The speed of each one in when they separate

Solution

a) Take the positive horizontal direction from left to right. Applying Newton's second law with the values ​​provided by the statement we have:

F ₂₁ = m ₁ to ₁

From where:

For the second skater:

b) The kinematic equations of uniformly accelerated rectilinear motion are used to calculate the velocity that they carry just as they separate:

The initial velocity is 0, since they were at rest in the middle of the track:

v _f = at

v _f1 = a ₁ t = -4 m / s ². 0.40 s = -1.6 m / s

v _f2 = a ₂ t = +2.5 m / s ². 0.40 s = +1 m / s

Results

As expected, person 1 being lighter acquires greater acceleration and therefore greater speed. Now notice the following about the product of the mass and the speed of each skater:

m ₁ v ₁ = 50 kg. (-1.6 m / s) = - 80 kg.m / s

m ₂ v ₂ = 80 kg. 1 m / s = +80 kg.m / s

The sum of both products is 0. The product of mass and speed is called momentum P. It is a vector with the same direction and sense of speed. When the skaters were at rest and their hands were in contact, it could be assumed that they formed the same object whose momentum was:

P _o = (m ₁ + m ₂) v _o = 0

After the push is completed, the amount of motion of the skating system remains 0. Therefore the amount of motion is conserved.

Examples of Newton's third law in everyday life

Walk

Walking is one of the most everyday actions that can be carried out. If observed carefully, the action of walking requires pushing the foot against the ground, so that it returns an equal and opposite force on the foot of the walker.

As we walk we constantly apply Newton's third law. Source: Pixabay.

It is precisely that force that allows people to walk. In flight, the birds exert force on the air and the air pushes the wings so that the bird propels itself forward.

Movement of a car

In a car, the wheels exert forces on the pavement. Thanks to the reaction of the pavement, it exerts forces on the tires that propel the car forward.

Sport

In sports, the forces of action and reaction are numerous and have a very active participation.

For example, let's see the athlete with his foot resting on a starter block. The block provides a normal force in reaction to the push that the athlete exerts on it. The result of this normal and the weight of the runner, result in a horizontal force that allows the athlete to propel himself forward.

The athlete uses the starter block to add forward momentum at the start. Source: Pixabay.

Fire hoses

Another example in which Newton's third law is present is in firefighters holding fire hoses. The end of these large hoses has a handle on the nozzle that the firefighter must hold when the jet of water exits, to avoid the recoil that occurs when the water is rushing out.

For the same reason, it is convenient to tie the boats to the dock before leaving them, because by pushing themselves to reach the dock, a force is provided to the boat that moves it away from it.

References

1. Giancoli, D. 2006. Physics: Principles with Applications. Sixth Edition. Prentice Hall. 80 - 82.
2. Rex, A. 2011. Fundamentals of Physics. Pearson. 73 - 75.
3. Tipler, P. 2010. Physics. Volume 1. 5th Edition. Editorial Reverté. 94 - 95.
4. Stern, D. 2002. From astronomers to spaceships. Taken from: pwg.gsfc.nasa.gov.