Mass — Question 7

Question 7

A zero-acceleration test cannot identify mass by division

A body of unknown positive finite mass is subject to zero net force and has zero acceleration. Use $F_{\mathrm{net}}=ma$.

1. Can these data determine m?

2. Why is $m=0/0$ not a measurement?

3. Design a second test that gives m, and evaluate it if 6 N produces 2 m/s$^2$.

Question 7 - Solution

A zero-acceleration test cannot identify mass by division

(a) Keep the equation before dividing.

$$0=m(0)\quad\text{for every }0<m<\infty$$ The equation is $0=m\cdot0$, true for every positive finite mass. It supplies no discrimination among masses.

(b) Recognize indeterminacy.

$$\frac00\text{ undefined},\qquad\{m:0=m\cdot0\}=(0,\infty)$$ Zero divided by zero is undefined. The observation does not imply zero mass or infinite mass; it simply lacks a nonzero response from which to infer a ratio.

(c) Apply a known nonzero resultant.

$$\boxed{m=\frac{F}{a}=\frac{6\,\mathrm N}{2\,\mathrm{m/s^2}}=3\,\mathrm{kg}}$$ For a nonzero calibrated force and measured acceleration, $m=F/a$. The proposed second test gives $6/2=3$ kg, provided force and acceleration are measured in an inertial frame and refer to the same body.