Physics Bootcamp

Subsection 6.2.1 Second Law for Fixed Mass

Consider an object of fixed mass, say a block of steel. When you apply a force on it, you find that its speed and/or direction of motion changes. This is what happens, for instance, when you let it fall or if you spin it in a circle. This is a universal behavior.

The amount of acceleration on different objects of same mass varies linearly with force applied.

When you apply same force on two objects of different masses, the resulting acceleration is inversely proportional to the mass.

These two relations imply

which can be turned into an equality

By choosing a unit of force, we can set \(k=1\text{.}\) Thus, we define a force to be 1 Newton(N) if it causes a one kg object to accelerate \(1\text{ m/s}^2\text{.}\) That is, if mass is in kg, acceleration in \(\text{m/s}^2\text{,}\) and force in \(\text{N}\text{,}\) then

These observations, along with experiments addressing acceleration when direction of velocity changes, e.g., in circular motion, were put together succinctly by Isaac Newton in the following vector equation, which is the expression of Newton's second law of motion for fixed-mass systems.

Note that this equation does not apply to systems that lose or gain mass during their motion, e.g., rocket (which loses mass) and rain drops (which gain mass). For these situations, we need a more general law, given next.

Subsection 6.2.2 What is mass?

We use second law of motion for fixed-mass, viz., \((\vec F = m\vec a)\) to deduce an empirical definition of mass, i.e., mass rather being some vague thing as “quantity of matter” will become an experimentally quantifiable quantity. We apply same force to two different objects and observe their resulting acceleration and deduce mass from the observations.

For instance, suppose we attach blocks of different masses to the same spring, stretching the spring to the same amount, and then releasing the blocks. We will find that their accelerations will be different.

The magnitudes of the resulting accelerations will be

We can divide out \(F\) to find that the ratio of their accelerations will be inversely proportional to their masses.

This equation lets us define mass operationally in relation to a reference mass as follows. Suppose we take a reference object and assign, arbitrarily, its mass to be one unit of mass, say \(1 \text{ kg}\text{.}\) After that, we perform the aforementioned experiment, i.e., apply an unknown but reproducible force on it. Suppose we find that the acceleration of this reference object is \(4\text{ m/s}^2\text{.}\)

Now, we take another object whose mass we do not know yet. We subject this object to the same force and observe its acceleration. Suppose we find that its acceleration is \(8\text{ m/s}^2\text{.}\) Then, Eq. (6.2.7) can be used to deduce the mass of this second object in relation to the mass of the first object.

The example above gives us a process by which, after we choose a reference object of unit mass, we can do experiments to assign mass to all other objects. We call this mass inertial mass.