Newton's Second Law for a System of Particles — Question 8

Question 8

Only one component is externally balanced

Two interacting particles have total mass $3.0\,\mathrm{kg}$. Their total external force is $(0,-6.0)\,\mathrm N$. Initially their center velocity is $(4,1)\,\mathrm{m/s}$. The force stays constant for $2.0\,\mathrm s$.

1. Find the center velocity after $2.0\,\mathrm s$.

2. Which velocity component is constant, and can internal forces make the other component constant?

Question 8 - Solution

Only one component is externally balanced

(a) Derivation and calculation

For the fixed total mass, $$\vec a_C=\frac{\vec F_{\mathrm{ext}}}M=\frac{(0,-6)}3=(0,-2)\ \mathrm{m/s^2}.$$ $$\boxed{\vec V_C(2)=(4,1)+(0,-2)(2)=(4,-3)\ \mathrm{m/s}}.$$

(b) Result and interpretation

Apply the center law separately to each component: $$\dot V_{C,x}=0,\qquad\dot V_{C,y}=-2\ \mathrm{m/s^2}.$$ Thus $V_{C,x}=4$ m/s is constant. Internal forces cancel in the total and cannot make $V_{C,y}$ constant against the specified net external force.

Physical check

The vertical velocity change is $-4$ m/s, consistent with a $-12$ N s external impulse.