QJEEAdv07Ph4526: JEE Advanced 2007 Physics - Class 11 JEEAdv07Ph4526 Solution: A particle moves in the X - Y plane under the influence of a…

Given that,

$\overrightarrow{\mathrm{P}}(t)=\mathrm{A}[\hat{i} \cos (k t)-\hat{j} \sin (k t)]$

where, $\mathrm{A}$ and $k=$ constant

$\begin{aligned}\text { Force }\overrightarrow{\mathrm{F}}) & =\frac{d \overrightarrow{\mathrm{P}}}{d t}\left(\text { from Newton's } 2^{\text {nd }}\right. \text { law) } \\\\\Rightarrow \quad \overrightarrow{\mathrm{F}} & =\frac{d}{d t}[\mathrm{~A}(\hat{i} \cos (k t)-\hat{j} \sin (k t))] \\\\& =\mathrm{A} k[\hat{i} \cos (k t)-\hat{j} \sin (k t)]\end{aligned}$

By applying dot product between $\vec{F} \& \vec{P}$ we have

$\begin{aligned}& \overrightarrow{\mathrm{F}} \cdot \overrightarrow{\mathrm{P}}=0 \\\\& \Rightarrow|\vec{F}||\vec{P}| \cos \theta=0 \\\\& \Rightarrow \quad \cos \theta=0 \\\\& \text {Hence,} \theta=90^{\circ}\end{aligned}$