Mass — Question 3

Question 3

Combining masses changes a common acceleration response

Use the inertial calibration relation $\vec F_{\mathrm{net}}=m\vec a$ for a constant-mass body in an inertial frame. The same 12 N net force accelerates cart A at 4 m/s$^2$ and cart B at 6 m/s$^2$ in separate tests. They are then joined and accelerate together under a total external net force of 12 N. Ignore connector mass.

1. Find the two masses.

2. Find the combined acceleration.

3. Why is averaging the two separate accelerations not valid?

Question 3 - Solution

Combining masses changes a common acceleration response

(a) Infer each mass.

$$m_A=\frac{12}{4}=3\,\mathrm{kg},\quad m_B=\frac{12}{6}=2\,\mathrm{kg}$$ $m_A=12/4=3$ kg and $m_B=12/6=2$ kg.

(b) Add masses for the combined translating body.

$$\boxed{a_{AB}=\frac{12}{3+2}=2.4\,\mathrm{m/s^2}}$$ The combined mass is 5 kg, hence $a=12/5=2.4$ m/s$^2$.

(c) Keep the total force fixed.

$$\frac1{a_{AB}}=\frac{m_A+m_B}{F}=\frac1{a_A}+\frac1{a_B}$$ The combined force is only 12 N, not 12 N on each part. Arithmetic mean acceleration, 5 m/s$^2$, has no basis here. In fact $1/a_{AB}=1/a_A+1/a_B$ for these same-force tests.