📂Classical Mechanics

Physics: The Definition of Mass, Force, and Momentum

Mass¹

In Newton’s laws of motion, inertia is described as the property that resists changes in motion. That is, the greater the inertia, the harder it is to move, and the smaller the inertia, the easier it is to move. This exactly aligns with our experience that it’s harder to push a heavier object than a lighter one. Hence, the magnitude of inertia can be expressed by the magnitude of mass. Mass refers to how heavy or light an object is. This defines the meaning of mass. Below explains how to define the value of mass.

Let’s assume there are two objects. Let’s call their masses $m_{1}$ and $m_{2}$, respectively. And let’s say they are moved in opposite directions by the same force. To put it simply, imagine a situation where a spring is nestled between the two objects, pressed and then released from both sides. The two objects are propelled at speeds $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$, respectively. At this point, the ratio of the masses of the two objects is defined as follows.

$$ \frac{m_{2}}{m_{1}}=\left|\frac{\mathbf{v}_{1}}{\mathbf{v}_{2}} \right| $$

By taking the mass $m_{1}$ of object 1 as the standard, the masses of other objects (materials) can be determined.

Momentum and Force

The product of an object’s mass and velocity is called momentum, written as $\mathbf{p}$. To distinguish it from angular momentum, it is also referred to as linear momentum.

$$ \mathbf{p}=m\mathbf{v} $$

Momentum, as the name suggests, is a physical quantity that a moving object has. Therefore, a change in an object’s motion means that the object’s momentum has increased or decreased. Then, the change in motion mentioned in Newton’s second law can be said to be the change in momentum over time. Furthermore, since force is said to change the motion of an object, Newton’s second law can be expressed as the following equation from the definition of momentum.

$$ \begin{equation} \mathbf{F}=k\frac{ d \mathbf{p}}{ d t } \end{equation} $$

In other words, ’the force $\mathbf{F}$ applied to an object is proportional to the change in the object’s momentum’. Here, $k$ is the proportionality constant. Assuming that the mass $m$ of the object does not change over time (as is the case from high school to college physics in many situations), the above equation can be written as follows.

$$ \mathbf{F}=k\frac{d(m\mathbf{v})}{dt}=km\frac{d \mathbf{v}}{dt}=km\mathbf{a} $$

Here, $\mathbf{a}$ is the acceleration that an object with mass $m$ has when force $\mathbf{F}$ is applied to it. If the proportionality constant is $k=1$, it becomes that famous equation.

$$ \mathbf{F}=m\mathbf{a} $$

From Newton’s laws of motion and the above definitions, the conservation of momentum principle naturally follows. The left side of $(1)$ is the net force acting on the system (or particle system), and the right side is the rate of change of the system’s momentum. If there is no external force, the rate of change of momentum is $0$, thus indicating that momentum is conserved.