Mass — Question 6

Question 6

One scalar mass must fit every vector component

Use the inertial calibration relation $\vec F_{\mathrm{net}}=m\vec a$ for a constant-mass body in an inertial frame. A measured net force is $(6,8)$ N and acceleration is $(1.5,2)$ m/s$^2$. A second report changes only $a_y$ to 1 m/s$^2$.

1. Find mass from the original report.

2. Test the second report for a single positive scalar mass.

3. Why is using only the ratio of vector magnitudes insufficient for consistency?

Question 6 - Solution

One scalar mass must fit every vector component

(a) Check each nonzero component.

$$m_x=\frac6{1.5}=4,\quad m_y=\frac82=4\ \Rightarrow\ \boxed{m=4\,\mathrm{kg}}$$ $6/1.5=8/2=4$ kg. The force and acceleration are parallel, and the magnitude ratio $10/2.5$ agrees.

(b) Compare independent component estimates.

$$m_x'=4\,\mathrm{kg}\ne m_y'=8/1=8\,\mathrm{kg}$$ The altered report gives 4 kg from x and 8 kg from y. No single mass satisfies both equations.

(c) Retain vector direction.

$$F_xa_y-F_ya_x=0\ \text{required};\quad6(1)-8(1.5)=-6\ne0$$ A magnitude ratio can always yield a number when acceleration is nonzero, even for nonparallel vectors. A scalar mass multiplies every component equally; both direction and proportionality must hold.