Solving Vector Equation: Obtain m from F=ma

[Mr Davis 97]

I have a question. Given a vector equation such as F = ma, how can we obtain a general expression for m, the mass? If the equation was scalar, this could easily be done by dividing F by a; however, we are dealing with vectors, and, to my knowledge, a vector divided by another vector is not defined in vector algebra. Therefore, how can we obtain a general expression for m?

 

[axmls]

You could always just write $$\mathbf{F}=m \mathbf{a} \implies \mathbf{F} \cdot \mathbf{a} = m \mathbf{a} \cdot \mathbf{a} \implies \mathbf{F} \cdot \mathbf{a} = m a^2 \implies m = \frac{\mathbf{F} \cdot \mathbf{a}}{a^2}.$$

But the acceleration and the net force are always in the same direction. So the dot product is just the scalar product of the magnitudes.

 

[pbuk]

Take any component of F and a and you have $F_{x_i} = m a_{x_i}$.

 

[jbriggs444]

Take any component of F and a and you have $F_{x_i} = m a_{x_i}$.

Either pick a non-zero component or, better yet, divide the magnitudes. As long as F and a are collinear (as they must be), this will give the right answer.

 

[tommyxu3]

For the directions of $F$ and $a$ are the same, we can get
$$\mathbf{F}=m\mathbf{a}\Rightarrow F\hat{e}=ma\hat{e}\Rightarrow F=ma \Rightarrow m=\frac{F}{a}.$$

 

[HallsofIvy]

For the directions of $F$ and $a$ are the same, we can get
$$\mathbf{F}=m\mathbf{a}\Rightarrow F\hat{e}=ma\hat{e}\Rightarrow F=ma \Rightarrow m=\frac{F}{a}.$$

That last would be better written $\frac{|F|}{|a|}$

 

[tommyxu3]

Sorry I used $\mathbf{F}$ as the vector force and $F$ as the magnitude of the force.

 

[HallsofIvy]

Oh, I see. So you were right all along!