Newton's Second Law for a System of Particles — Question 5

Question 5

Inferring a hidden external force

A $5.0\,\mathrm{kg}$ fixed-mass system has center coordinate $X(t)=1+2t+0.40t^3$ in SI units. A known external force $F_A=4.0\,\mathrm N$ acts along $+x$; one other external horizontal force acts.

1. Find the other force as a function of time and its value at $t=2.0\,\mathrm s$.

2. At what time is the total external force zero? Does the center then have to be at rest?

Question 5 - Solution

Inferring a hidden external force

(a) Derivation and calculation

Differentiate the measured trajectory twice: $$V_C=2+1.20t^2,\qquad a_C=2.40t,\qquad F_{\mathrm{ext}}=5(2.40t)=12t.$$ $$\boxed{F_B=F_{\mathrm{ext}}-F_A=12t-4\ \mathrm N},\qquad \boxed{F_B(2)=20\ \mathrm N},$$ where numerical time is in seconds.

(b) Result and interpretation

The net-force zero and corresponding center velocity are $$12t=0\Rightarrow\boxed{t=0},\qquad\boxed{V_C(0)=2.0\ \mathrm{m/s}}.$$ The center is moving even though its instantaneous acceleration is zero.

Physical check

At zero time the two external forces are $+4$ N and $-4$ N, so they balance.