Newton's Second Law — Question 7

Question 7

An accelerating coordinate system changes the equation’s form

A 2 kg particle has physical horizontal net force +6 N in inertial frame G. Observer B accelerates at +1 m/s$^2$ relative to G with nonrotating parallel axes.

1. Find the particle’s acceleration in G and relative to B.

2. Compare $m a_{P/B}$ with the physical force.

3. What extra effective term would make the equation work in B?

Question 7 - Solution

An accelerating coordinate system changes the equation’s form

(a) Apply the law in the inertial frame first.

$$a_{P/G}=6/2=3,\quad\boxed{a_{P/B}=a_{P/G}-a_{B/G}=3-1=2\,\mathrm{m/s^2}}$$ $a_{P/G}=6/2=3$ m/s$^2$. Relative acceleration is $a_{P/B}=3-1=2$ m/s$^2$.

(b) Identify the mismatch.

$$ma_{P/B}=2(2)=4\,\mathrm N\ne F_{\mathrm{physical}}=6\,\mathrm N$$ $ma_{P/B}=4$ N, whereas the physical net force is 6 N. Substituting an accelerating observer’s coordinate acceleration directly into the inertial equation is invalid.

(c) Subtract the frame acceleration term.

$$ma_{P/B}=F_{\mathrm{physical}}-ma_{B/G}=6-2(1)=4\,\mathrm N$$ $m a_{P/B}=F_{\mathrm{physical}}-m a_{B/G}=6-2=4$ N. The effective $-2$ N term arises from the chosen coordinates, not a newly identified physical agent acting on the particle.