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(D) Given mass $m = 1 \, kg$ and force $\vec{F} = 2 \sin(3 \pi t) \hat{i} + 3 \cos(3 \pi t) \hat{j}$.Acceleration $\vec{a} = \frac{\vec{F}}{m} = 2 \si

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(D) Given mass $m = 1 \, kg$ and force $\vec{F} = 2 \sin(3 \pi t) \hat{i} + 3 \cos(3 \pi t) \hat{j}$.Acceleration $\vec{a} = \frac{\vec{F}}{m} = 2 \sin(3 \pi t) \hat{i} + 3 \cos(3 \pi t) \hat{j}$.Velocity $\vec{v} = \int \vec{a} dt = \int (2 \sin(3 \pi t) \hat{i} + 3 \cos(3 \pi t) \hat{j}) dt = -\frac{2}{3\pi} \cos(3\pi t) \hat{i} + \frac{3}{3\pi} \sin(3\pi t) \hat{j} + \vec{C}_1$.At $t=0$,$\vec{v}=0$,so $0 = -\frac{2}{3\pi} \hat{i} + \vec{C}_1 \implies \vec{C}_1 = \frac{2}{3\pi} \hat{i}$.Thus,$\vec{v} = \left( \frac{2}{3\pi} - \frac{2}{3\pi} \cos(3\pi t) \right) \hat{i} + \frac{1}{\pi} \sin(3\pi t) \hat{j}$.Position $\vec{r} = \int \vec{v} dt = \left( \frac{2t}{3\pi} - \frac{2}{9\pi^2} \sin(3\pi t) \right) \hat{i} - \frac{1}{3\pi^2} \cos(3\pi t) \hat{j} + \vec{C}_2$.At $t=0$,$\vec{r}=0$,so $0 = 0 \hat{i} - \frac{1}{3\pi^2} \hat{j} + \vec{C}_2 \implies \vec{C}_2 = \frac{1}{3\pi^2} \hat{j}$.The position vector is $\vec{r} = \left( \frac{2t}{3\pi} - \frac{2}{9\pi^2} \sin(3\pi t) \right) \hat{i} + \left( \frac{1}{3\pi^2} - \frac{1}{3\pi^2} \cos(3\pi t) \right) \hat{j}$.At $t=1 \, s$,$\vec{r}(1) = \left( \frac{2}{3\pi} - 0 \right) \hat{i} + \left( \frac{1}{3\pi^2} - \frac{1}{3\pi^2} (-1) \right) \hat{j} = \frac{2}{3\pi} \hat{i} + \frac{2}{3\pi^2} \hat{j}$.Since this result is not in the options,the correct choice is $D$.

$A$ body of mass $1 \, kg$ is acted upon by a force $\vec F = 2\sin(3\pi t)\hat i + 3\cos(3\pi t)\hat j$. Find its position at $t = 1 \, s$ if at $t = 0$ it is at rest at the origin.

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