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A particle of mass $M=0.2 kg$ is initially at rest in the $x y$-plane at a point $( x =-l, y =-h)$, where $l=10 m$ and $h=1 m$. The particle is accelerated at time $t =0$ with a constant acceleration $a =10 m / s ^2$ along the positive $x$-direction. Its angular momentum and torque with respect to the origin, in SI units, are represented by $\vec{L}$ and $\vec{\tau}$, respectively. $\hat{i}, \hat{j}$ and $\hat{k}$ are unit vectors along the positive $x , y$ and $z$-directions, respectively. If $\hat{k}=\hat{i} \times \hat{j}$ then which of the following statement($s$) is(are) correct?
$(A)$ The particle arrives at the point $(x=l, y=-h)$ at time $t =2 s$.
$(B)$ $\vec{\tau}=2 \hat{ k }$ when the particle passes through the point $(x=l, y=-h)$
$(C)$ $\overrightarrow{ L }=4 \hat{ k }$ when the particle passes through the point $(x=l, y=-h)$
$(D)$ $\vec{\tau}=\hat{ k }$ when the particle passes through the point $(x=0, y=-h)$
$A,B,D$
$A,B,C$
$A,B$
$A,D$
Solution
$\overrightarrow{ r }_{ A }=-\hat{ j }$
$S =\frac{1}{2} a t ^2$
$20=\frac{1}{2} \times 10 \times t ^2$
$t =2 sec$
$\vec{\tau}_0=\overrightarrow{ r } \times \overrightarrow{ F } ; \overrightarrow{ r }_{ B }=10 \hat{ i }-\hat{ j }$
$\overrightarrow{ F }= ma =0.2 \times 10 \hat{ i }=2 \hat{ i }$
$\vec{\tau}_0=(10 \hat{ i }-\hat{ j }) \times(2 \hat{ i })$
$\vec{\tau}_0=2 \hat{ k }$
$\overrightarrow{ L }_0=\overrightarrow{ r }_{ B } \times \overrightarrow{ p }=\overrightarrow{ r }_{ B } \times m \overrightarrow{ v }$
$\overrightarrow{ v }^2=\overrightarrow{ a }=10 \hat{ i } \times 2=20 \hat{ i }$
$\overrightarrow{ L }_0=(0.2)[(10 \hat{ i }-\hat{ j }) \times 20 \hat{ i }]=4 \hat{ k }$
At point $A (0,-1)$
$\vec{\tau}_0=\vec{r}_{ A } \times \overrightarrow{ F }=(-\hat{ j }) \times 2 \hat{ i }=2 \hat{ k }$
