એક કણ \(x-y\) સમતલમાં બળ \(\vec{F}\) ની અસર હેઠળ ગતિ કરે છે જેથી તેનો રેખીય વેગ \(\vec{P}(t)=\hat{i} \cos (k t)-\hat{j} \sin (k t)\) છે. જો \(k\) અચળ હોય, તો \(\overrightarrow{\mathrm{F}}\) અને \(\overrightarrow{\mathrm{P}}\) વચ્ચેનો કોણ _______ હશે.

\(\because \overrightarrow{\mathrm{P}}=\mathrm{m} \overrightarrow{\mathrm{v}}\)

\(\therefore \hat{\mathrm{P}}=\hat{\mathrm{v}}\)

\(\Rightarrow \hat{\mathrm{v}}=\cos (\mathrm{kt}) \hat{\mathrm{i}}-\sin (\mathrm{kt}) \hat{\mathrm{j}}\)

\(\hat{\mathrm{a}}=\frac{-\mathrm{k} \sin (\mathrm{kt}) \hat{\mathrm{i}}-\mathrm{k} \cos (\mathrm{kt}) \hat{\mathrm{j}}}{\mathrm{k}}\)

\(\Rightarrow \hat{\mathrm{a}}=-\sin k t \hat{\mathrm{i}}-\cos k \hat{\mathrm{t}}\)

\(\because \hat{\mathrm{F}}=\hat{\mathrm{a}}=-\sin \mathrm{kt} \hat{\mathrm{i}}-\cos k t \hat{\mathrm{j}}\)

\(\cos \theta=\frac{\hat{\mathrm{F}} \cdot \hat{\mathrm{P}}}{|\hat{\mathrm{F}}| \hat{\mathrm{P}} \mid}=-\frac{\sin k t \cos \mathrm{t}+\sin k t \cos \mathrm{t}}{1 \times 1}=0\)

\(\Rightarrow \theta=\frac{\pi}{2}\)