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Introduction to Newton's Laws of Motion

Newton's Second Law of Motion

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Introduction to Newton's Laws of Motion

Newton's Second Law of Motion

Andrew Zimmerman Jones
The acceleration produced by a particular force acting on a body is directly proportional to the magnitude of the force and inversely proportional to the mass of the body.
- Newton's Second Law of Motion, translated from the Principia's Latin
The mathematical formulation of the second law is shown to the right, with F representing the force, m representing the object's mass and a representing the object's acceleration.

This formula is extremely useful in classical mechanics, as it provides a means of translating directly between the acceleration of and force acting upon a given mass. A large portion of classical mechanics ultimately breaks down to applying this formula in different contexts.

The sigma symbol to the left of the force indicates that it is the net force, or the sum of all the forces, that we are interested in. As vector quantities, the direction of the net force will also be the same direction as the acceleration. You can also break the equation down into x & y (and even z) coordinates, which can make many elaborate problems more manageable, especially if you orient your coordinate system properly.

You'll note that when the net forces on an object sum up to zero, we achieve the state defined in Newton's First Law - the net acceleration must be zero. We know this because all object have mass (in classical mechanics, at least). If the object is already moving it will continue to move at a constant velocity, but that velocity will not change until a net force is introduced. Obviously, an object at rest will not move at all without a net force.

The Second Law in Action

A box with a mass of 40 kg sits at rest on a frictionless tile floor. With your foot, you apply a 20 N force in a horizontal direction. What is the acceleration of the box?
The object is at rest, so there is no net force except for the force your foot is applying. Friction is eliminated. Also, there's only one direction of force to worry about. So this problem is very straightforward.

You begin the problem by defining your coordinate system. In this case, that's easy - the +x direction will be the direction of the force (and, therefore, the direction of the acceleration). The mathematics is similarly straightforward:

F = m * a

F / m = a

20 N / 40 kg = a = 0.5 m / s2

The problems based on this law are literally endless, using the formula to determine any of the three values when you are given the other two. As systems become more complex, you will learn to apply frictional forces, gravity, electromagnetic forces, and other applicable forces to the same basic formula.

Newton's Three Laws of Motion

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