Newton’s Laws of Motion

In 1687 Isaac Newton published his Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) wherein he stated three fundamental physical laws that describe how a body reacts to forces acting upon it. These laws are extremely important as they form the basis for classical mechanics.

There are variations in the wording of the laws; this is one such variation that I tend to use.

Newton’s First Law: A body will remain stationary or moving at a constant velocity unless acted upon by a resultant force.

The idea that a body will remain at rest unless acted upon by a force can be demonstrated by any object at rest, say a mug sitting on a table. The mug won’t spontaneously jump up into the air, or fall through the table, if no resultant forces act upon it. Pushing it, or exploding it, requires a force.

The idea that a body will remain moving at a constant velocity unless acted upon by a force can be demonstrated by objects floating in outer space. You may have witnessed an astronaut releasing an object from their grasp, and the object moves continually in one direction at a constant speed. It continues moving in that direction at that speed because no forces act upon it (whereas gravity is significant here on Earth). Only when a force acts upon it, for example the astronaut taking hold of the object again, will the object change speed or direction. This idea can’t be demonstrated on the surface of the Earth due to the gravity of the Earth acting on every object, and also due to air resistance. In outer space these two effects still apply but are negligible.

Newton’s Second Law: The resultant force acting upon a body is equal to the rate of change of momentum of that body.

This law can be shown by the equation:

Newton's Second Law

If the mass is constant, this equation can be simplified to the more famous:

Newton's Second Law (Constant Mass)

This equation, that force is the product of mass and acceleration, is one that many people will recognise as Newton’s second law, but it only applies if the mass does not change. This equation would not be able to be used in, for instance, the context of a rocket burning up fuel.

Newton’s Third Law: When the first body exerts a force on the second body, the second body exerts an equal and opposite force on the first body.

Picture the mug sitting on the table again. What is the vertical pair of forces in this system? If you said the weight of the mug downwards on the table and the reaction force of the table upwards on the mug, you have made a classic rookie mistake. Pairs of forces in terms of Newton’s third law must:

  • be of equal magnitude
  • be of the same type of force
  • act in opposite directions
  • act on different bodies

The weight of the mug and the reaction force of the table are equal in magnitude, and do act in opposite directions, but they are not the same type of force (for example electromagnetic or gravitational), and they both act on the mug. An example of a pair of forces would be the gravitational force of the Earth on the mug, and the gravitational force of the mug on the Earth.

A common example used to demonstrate Newton’s third law is when you walk. Your foot pushes downwards into the ground, while the ground pushes up onto your foot, allowing you to move forwards. They are equal in magnitude, are both mechanical forces, are acting in opposite directions and act on different bodies, so therefore they are a pair.



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