Mass — Question 4

Question 4

A slope determines mass only when force and acceleration are matched

A cart’s measured pairs $(a,F_{\mathrm{net}})$ are $(1,2.5)$, $(2,5.0)$ and $(4,10.0)$ in SI units. Assume constant mass and inertial measurements.

1. Find the mass from the slope of F versus a.

2. Predict the net force for $a=-3$ m/s$^2$.

3. If a graph instead plots a vertically against F horizontally, what is its slope?

Question 4 - Solution

A slope determines mass only when force and acceleration are matched

(a) Use the proportional relation.

$$m=\frac{\Delta F}{\Delta a}=\frac{10-2.5}{4-1}=\boxed{2.5\,\mathrm{kg}}$$ $F_{\mathrm{net}}/a=2.5$ kg in every pair. The F-versus-a slope is the mass.

(b) Keep the acceleration sign.

$$F(-3)=2.5(-3)=-7.5\,\mathrm N$$ $F_{\mathrm{net}}=2.5(-3)=-7.5$ N; negative force gives negative acceleration along the chosen axis.

(c) Invert the axes correctly.

$$\frac{\Delta a}{\Delta F}=\frac1m=0.4\,\mathrm{kg^{-1}}$$ The a-versus-F slope is $1/m=0.4$ kg$^{-1}$, equivalently $(\mathrm{m/s^2})/\mathrm N$. Calling that slope the mass would invert the physical relation.