Question

The momentum of a moving particle is vectorially given as $\overrightarrow{p}=p_0$ $(cost\omega \hat{i}+sint\omega \hat{j})$, where $t$ stands for time. Choose the correct option.

• A. the applied force is constant.
• B. the momentum is constant.
• C. the applied force always remains perpendicular to the momentum.
• D. the applied force is always parallel to the momentum.

Answer 1

Answer: (C)

the applied force always remains perpendicular to the momentum.

$\overrightarrow{P}=P_{0}cos\omega t\hat{i}+P_{0}sin\omega t\hat{j}\frac{d\overrightarrow{P}}{dt}=\frac{d(P_{0}cos\omega t\hat{i}+P_{0}sin\omega t\hat{j})}{dt}\Rightarrow \frac{d\overrightarrow{P}}{dt}=-P_0\omega sin\omega t\hat{i}+P_0\omega cos\omega t\hat{j}\Rightarrow \overrightarrow{F}=-P_0\omega sin\omega t\hat{i}+P_0\omega cos\omega t\hat{j}\Rightarrow \overrightarrow{F}\cdot \overrightarrow{P}=(-P_0\omega sin\omega t\hat{i}+P_0\omega cos\omega t\hat{j})\cdot (P_{0}cos\omega t\hat{i}+P_{0}sin\omega t\hat{j})\Rightarrow \overrightarrow{F}\cdot \overrightarrow{P}=-{P_0}^2\omega sin\omega tcos\omega t+{P_0}^2\omega sin\omega tcos\omega t=0\Rightarrow \overrightarrow{F}isperpendicularto\overrightarrow{P}$