📂Physics

Mass

Definition¹

Mass is one of the fundamental properties of a body and represents the magnitude of its inertia. The larger the mass, the more difficult it is to change the body’s state of motion.

Description

Mass is a scalar physical quantity and has a value greater than or equal to $0$.

$$ m \ge 0 $$

We have the everyday experience that pushing a heavy object is harder than pushing a light one. In other words, heavy objects are difficult to set into motion, and we say such objects “have large mass.” Thus mass can be regarded as a measure of how heavy or light an object is. However, here “heavy” is a colloquial expression; in physics it refers to the degree of resistance to changes in motion, i.e., the magnitude of inertia. In Newton’s laws of motion, inertia is described as the property that resists changes in motion. If inertia is large, it is difficult to change the motion; if inertia is small, it is easy to change the motion. The mass in this sense is called the inertial mass. Newton’s second law holds for the inertial mass $m$.

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

One method to determine values of mass is to use the interaction between two bodies. Suppose there are two bodies. Let the masses of each body be $m_{1}$ and $m_{2}$. And suppose the two bodies exert forces of equal magnitude on each other and move in opposite directions. In simple terms, imagine inserting a spring between the two bodies, compressing it on both sides, and then releasing it. The two bodies are ejected with velocities $\mathbf{v}_{1}$ and $\mathbf{v}_{2}$, respectively. At this time, the ratio of the masses of the two bodies is defined as follows.

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

If we take the mass of object 1 $m_{1}$ as a reference, we can determine the masses of other objects (materials).

Weight

In everyday usage, the quantity weight is used more often than mass. Mass is an intrinsic property of the object itself, whereas weight is the force the object experiences due to gravity. For example, the mass of the same object on Earth and on the Moon does not change, but because lunar gravity is weaker than Earth’s, the weight is smaller. The weight of an object $W$ can be expressed using the gravitational acceleration $g$ as follows.

$$ W = mg $$

Saying colloquially “I weigh 60 kilograms” is strictly inaccurate; one should say “I have a mass of 60 kilograms” or “I weigh 588 newtons.” Of course, saying that in everyday conversation or correcting others for that phrasing may lead to social ostracism.

Unit

In the International System of Units, the unit of mass is the kilogram, symbol $\mathrm{kg}$. In the past, the kilogram was defined based on a specific physical artifact called the prototype kilogram, and $1\mathrm{kg}$ was defined accordingly. However, in modern practice the kilogram is defined based on fundamental constants. Currently the kilogram is defined by fixing the numerical value of the Planck constant $h$ exactly. With the second and the metre defined, the kilogram is defined such that the numerical value of the Planck constant expressed in the unit $\mathrm{kg \cdot m^2 \cdot s^{-1}}$ is exactly as follows.

$$ h = 6.62607015 \times 10^{-34} \mathrm{kg \cdot m^2 \cdot s^{-1}} $$