While trying to understand the second law of Newton from "An Introduction to Mechanics" by Kleppner and Kolenkow, I came across the following lines that I don't understand:
"It is natural to assume that for three-dimensional motion, force, like acceleration, behaves like a vector. Although this turns out to be the case, it is not obviously true. For instance, if mass were different in different directions, acceleration would not be parallel to force and force and acceleration could not be related by a simple vector equation. Although the concept of mass having different values in different directions might sound absurd, it is not impossible. In fact, physicists have carried out very sensitive tests on this hypothesis, without finding any variation. So, we can treat mass as a scalar, i.e. a simple number, and write $\vec{F} = m\vec{a}$."
The lines above lead me to question:
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Why is it not" obviously true" that force behaves like a vector?
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Why is it not impossible for mass values to be different in different directions?
Best Answer
I think that an example will clear out your doubts.
Consider a 3D system in which you have three axis $xyz$. Consider a force that can be written as: $$ \mathbf{F} = F\hat{x} + F\hat{y} +F\hat{z} $$ Therefore, this force is identical in each direction. If we have three different masses depending on the direction we are considering: $$ m_x, m_y, m_z$$ Then we will have different accelerations depending on the direction in which we are studying the motion: $$ a_x = F/m_x, a_y = F/m_y, a_z = F/m_z$$ And therefore, the vector acceleration $\mathbf{a} $ will not be parallel to the vector force $\mathbf{F} $, since in each direction the masses (and therefore, the components of the acceleration) are different.
However, this has shown up to be wrong, according to some experiment. Therefore, the book is saying you that there is an evidence that the mass is identical regardless of the spatial direction you are considering.