8.2 Newton's second law of motion

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Second law of motion is the centerpiece of classical dynamics, providing exact connection between force (cause) and acceleration (effect).

The second law of motion determines the effect of net force on a body. The first law only defines the natural state of the motion of a body, when net force on the body is zero. It does not provide us with any tool to quantitatively relate force and acceleration (rate of change in velocity).

Second law of motion is the centerpiece of classical dynamics as it states the exact relation between force (cause) and acceleration (effect). This law has an explicit mathematical form and, therefore, has the advantage of quantitative measurement. As a matter of fact, the only available quantitative definition of force is given in terms of second law : “Force is equal to acceleration produced in unit mass.”

It must be clearly understood that the three laws of motion could well have been replaced by this single law of motion. However, the three laws are presented as they are, because first and third laws convey fundamental nature of "motion" and "force" which are needed to complete our understanding about them.

The second law of motion is stated in terms of linear momentum. It would, therefore, be appropriate that we first familiarize ourselves with this term.

Linear momentum

Linear momentum of a particle is defined as a vector quantity, having both magnitude and direction. It is the product of mass (a scalar quantity) and velocity (a vector quantity) of a particle at a given instant.

p = m v

The dimensional formula of linear momentum is $\left[ML{T}^{-1}\right]$ and its SI unit of measurement is " $kg-m∕s$ ".

Few important aspects of linear momentum need our attention :

First, linear momentum is a product of positive scalar (mass) and a vector (velocity). It means that the linear momentum has the same direction as that of velocity.

Second, we have earlier referred that motion of a body is represented completely by velocity. But, the velocity alone does not convey anything about the inherent relation that “change in velocity” has with force. The product of mass and velocity in linear momentum provides this missing information.

In order to fully appreciate the connection between motion and force, we may consider two balls of different masses, moving at same velocity, which collide with a wall. It is our everyday common sense that tells us that the ball with greater mass exerts bigger force on the wall. We may, therefore, conclude that linear momentum i.e. the product of mass and velocity represents the “quantum of motion”, which can be connected to force.

It is this physical interpretation of linear momentum that explains why Newton’s second of motion is stated in terms of linear momentum as this quantity (not the velocity alone) connects motion with force.

Newton’s second law of motion

The second law of motion is stated differently. We have chosen to state the law as given here :

Newton’s second law of motion
The time rate of change of momentum of a body is equal to the net (resultant) external force acting on the body.

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