A) \[60{}^\circ \]
B) \[30{}^\circ \]
C) \[45{}^\circ \]
D) \[90{}^\circ \]
Correct Answer: D
Solution :
From Newtons second law of motion the force acting is defined as rate of change of momentum. \[F=\frac{d\overrightarrow{p}}{dt}\] Given, \[\overrightarrow{P}=3\cos 4t.\hat{i}+3\sin 4t.\,\hat{j}\] \[\overrightarrow{F}=\frac{d\overrightarrow{P}}{dt}=-12\,\sin 4t.\hat{i}+12\cos 4t.\hat{j}\] Also, \[\overrightarrow{F}.\overrightarrow{P}=|\overrightarrow{F}||\overrightarrow{P}|\cos \theta \] \[=(-12\sin 4t.\hat{i}+12\cos 4t.\hat{j}).(3\cos 4t.\hat{i}\] \[+3\sin 4t.\hat{j})\] \[=36(-\sin 4t\cos 4t+\cos 4t\sin 4t)=0\] \[\Rightarrow \] \[|\overrightarrow{F}|.|\overrightarrow{P}|\cos \theta =0\] Since, \[|\overrightarrow{F}|\ne 0,|\overrightarrow{P}|\ne 0\] \[\therefore \] \[\cos \theta =0\] Hence, \[\theta =90{}^\circ \]
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