Inverse Proportionality in $F=ma$ equation

Question

If two quantities are inversely related to each other (mass and acceleration), does that mean that they do not depend on each other, i.e. changing mass does not have any effect on acceleration?

My confusion stems from this explanation to a practice problem about the mass and acceleration of a block on a surface where the only force acting on the block is force friction:

"The acceleration of the block does not depend on its mass. The net force on the block directly depends on its mass, but the acceleration is proportional to the force and inversely proportional to the mass, so the acceleration does not depend on the mass"

Answer 1

You should be confused. Who wrote that? Anyway, we thankfully have mathematics to clear things up:

"The acceleration of the block does not depend on its mass."

OK. We'll see about that:

"The net force on the block directly depends on its mass,"

$$ F = \mu M^{\alpha} $$

where $\mu$ is some constant of propotionality. Presumably the adverb "directly" means "linearly", so we can set $\alpha=1$. Hence:

$$ F = \mu M $$

"but the acceleration is proportional to the force"

$$ a \propto F = \mu M $$

"and inversely proportional to the mass",

$$ a \propto \frac 1 M $$

which sounds likes:

$$ a \propto \frac 1 M \stackrel{?}{\propto} F \propto \mu M $$

meaning $a \stackrel{?}\propto M^2$,

...but that's wrong... I forgot something. They said "and", which is subtle, and means that that step should be:

$$ a \propto F \times \frac 1 M = \frac F M = \frac{\mu M}M = \mu$$

"so the acceleration does not depend on the mass"

Correct.

Answer 2

Your confusion stems from trying to generalise the specific statement that applies only to the practice problem.

First, as noted in JEB's answer, the confusing statement:

"The acceleration of the block does not depend on its mass. The net force on the block directly depends on its mass, but the acceleration is proportional to the force and inversely proportional to the mass, so the acceleration does not depend on the mass"

can be much simply written as:

The acceleration of the block does not depend on its mass, because $F=ma$.

Taken by itself, the above is clearly nonsense. But the problem specifies:

...a practice problem about the mass and acceleration of a block on a surface where the only force acting on the block is force friction...

Importantly, the part in bold tells you that $F=\mu mg$, where $\mu$ is the friction coefficient. So: \begin{equation} F=\mu mg = ma \quad\implies\quad a = \mu g\,, \end{equation} which is indeed independent of the mass.