Mass — Question 1

Question 1

Equal applied net forces compare inertial masses

Use the inertial calibration relation $\vec F_{\mathrm{net}}=m\vec a$ for a constant-mass body in an inertial frame. An identical net force gives reference body R, of mass 2 kg, acceleration 3 m/s$^2$ and body B acceleration 1.2 m/s$^2$.

1. Find the force and the mass of B.

2. Find $m_B/m_R$ directly from the acceleration ratio.

3. Why must the net forces, rather than merely two hand pushes, be equal?

Question 1 - Solution

Equal applied net forces compare inertial masses

(a) Calibrate with the reference.

$$F=m_Ra_R=2(3)=6\,\mathrm N,\quad\boxed{m_B=\frac6{1.2}=5\,\mathrm{kg}}$$ $F=2(3)=6$ N, so $m_B=6/1.2=5$ kg.

(b) Invert the response ratio.

$$m_Ra_R=m_Ba_B\ \Rightarrow\ \frac{m_B}{m_R}=\frac{a_R}{a_B}=\frac3{1.2}=2.5$$ $m_B/m_R=a_R/a_B=3/1.2=2.5$. The larger mass has the smaller acceleration under the same force.

(c) Account for all interactions.

$$m=\frac{P-R}{a},\qquad P_A=P_B\ \not\Rightarrow\ P_A-R_A=P_B-R_B$$ Equal applied pushes do not guarantee equal resultant forces if resistance or other forces differ. The mass comparison is valid only when the full force sums are equal or the other contributions are measured.