Question

The linear momentum of a particle is given by $P=(a\sin t\hat{i}-a\cos t\hat{j})$ kg-m/s A force $\overrightarrow{F}$ is acting on the particle Select correct alternative/s

• A. Linear momentum $\overrightarrow{P}$ of particle is always parallel to $\overrightarrow{F}$
• B. Linear momentum $\overrightarrow{P}$ of particle is always perpendicular to $\overrightarrow{F}$
• C. Linear momentum $\overrightarrow{P}$ is always constant
• D. Magnitude of linear momentum is constant with respect to time

Answer 1

Answer: (B)

Linear momentum $\overrightarrow{P}$ of particle is always perpendicular to $\overrightarrow{F}$

From newton's second law of motion,

$\overrightarrow{F}=\frac{\overrightarrow{dP}}{dt}$

$\overrightarrow{F}=\frac{d(asint\hat{i}-acost\hat{j})}{dt}$

$\overrightarrow{F}=acost\hat{i}+asint\hat{j}$

Clearly, it is not constant or parallel with the momentum vector.

Finding dot product of force and momentum vectors to find the angle between them,

$\overrightarrow{F}.\overrightarrow{P}=|\overrightarrow{F}||\overrightarrow{P}|\cos \theta$

$a^2\sin t\cos t-a^2\sin t\cos t=a^{2}cos\theta$

$cos\theta =0$

Hence, force and momentum vectors are perpendicular to each other.