The Linear Momentum of a System of Particles — Question 5

Question 5

The derivative of a system momentum record

A fixed collection of particles has total mass $4\,\mathrm{kg}$ and $\vec P(t)=(8+2t^2,12-4t)\,\mathrm{kg\,m/s}$ with SI numerical time.

1. Find the external force at $t=2\,\mathrm s$.

2. Find the center displacement from $t=0$ to $t=2\,\mathrm s$.

Question 5 - Solution

The derivative of a system momentum record

(a) Derivation and calculation

Differentiate the total momentum of the fixed collection: $$\vec F_{\mathrm{ext}}(t)=\frac{d\vec P}{dt}=(4t,-4)\ \mathrm N.$$ $$\boxed{\vec F_{\mathrm{ext}}(2)=(8,-4)\ \mathrm N}.$$

(b) Result and interpretation

Divide momentum by total mass before integrating velocity: $$\vec V_C=\frac{\vec P}{4}=(2+t^2/2,3-t).$$ $$\Delta\vec R_C=\int_0^2\vec V_C\,dt =\left[2t+\frac{t^3}{6},3t-\frac{t^2}{2}\right]_0^2 =\boxed{(16/3,4)\ \mathrm m}.$$

Physical check

The force is the time derivative of momentum; displacement instead comes from its time integral divided by mass.