6.2: Defining mass

Definition: Mass

What is the mass of an object? Newton’s second law tells us that if there is a constant force \(\overrightarrow{\boldsymbol{F}}_{\text {ref }}\) acting on the object, and we measure an acceleration vector \(\overrightarrow{\boldsymbol{a}}_{i}\), the object’s mass is given by:

\[m_{i} \equiv \frac{\left|\overrightarrow{\boldsymbol{F}}_{\mathrm{ref}}\right|}{\left|\overrightarrow{\boldsymbol{a}}_{i}\right|} \tag{6.2} \label{6.2}\]

where we used the \(\equiv\) sign to indicate that this is the definition of the mass, or inertial mass, of the object.

The object’s mass obeys Equation 6.2 irrespective of the properties of the force \(\overrightarrow{\boldsymbol{F}}_{\text {ref }}\), and is thus an intrinsic property of the object that normally does not change. To assign a value to the mass we use the international systems of units, the SI units. The previous equation tells us that if we apply a force of \(1 \mathrm{~N}\) and observe an acceleration of \(1 \mathrm{~m} / \mathrm{s}^{2}\), the mass of the object is \(1 \mathrm{~kg}\).

Figure 6.1: Newton’s second law defines both the force vector \(\overrightarrow{\boldsymbol{F}}_{i}\) acting on a point mass and its mass \(m_{i}\).

6.2.1 Weight of a mass

Since measuring accelerations is not always easy, in practice we often determine the mass of an object via the gravitational force acting on the object, e.g. by using a weighing scale. This is possible because it was experimentally determined that the gravitational force on earth is proportional to the mass of the object: \(\overrightarrow{\boldsymbol{F}}_{i, g}=m_{i} \overrightarrow{\boldsymbol{g}}\). The magnitude of the gravitational force on the object on earth’s surface is defined as the weight \(W\) of the object, so we can write:

\[W_{i} \equiv m_{i}|\overrightarrow{\boldsymbol{g}}| \tag{6.3} \label{6.3}\]

Here, the gravitational acceleration vector \(\overrightarrow{\boldsymbol{g}}\) has a magnitude of approximately \(9.8 \mathrm{~m} / \mathrm{s}^{2}\) on earth and points towards the center of the earth. The gravitational mass of an object can be determined using the equation \(m_{i}=W_{i} /|\overrightarrow{\boldsymbol{g}}|\) and has experimentally been found to be equal to its inertial mass as determined from Newton’s second law.

6.2.2 Point mass

Concept: Point mass

A point mass \(i\) is an object that behaves like a mathematical point in space at position vector \(\overrightarrow{\boldsymbol{r}}_{i}\) and has mass \(m_{i}\).

Since it is located at a single point, a point mass can only have a single position, velocity and acceleration vector. The point mass is an important concept, because Newton’s laws strictly only hold for point masses. Moreover, the smallest known particles, the elementary particles, behave very similar to point masses, and all larger objects are made out of elementary particles. Also, as we will see, the dynamics of larger objects can often still be well approximated by treating them as a point mass, therefore the term point mass is also used for larger objects that move like a point mass, e.g. because they do not rotate.