Question

A particle of mass $M=0.2kg$ is initially at rest in the $xy$-plane at a point $(x=-\ell ,y=-h)$, where $\ell =10m$ and $h=1m$. The particle is accelerated at time $t=0$ with a constant acceleration $a=10m/s^2$ 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 $(x=\ell ,y=-h)$ at time $t=2s$.
• B. $\vec{\tau }=2\hat{k}$ when the particle passes through the point $(x=\ell ,y=-h)$
• C. $\vec{L}=4\hat{k}$ when the particle passes through the point $(x=\ell ,y=-h)$
• D. $\vec{\tau }=\hat{k}$ when the particle passes through the point $(x=0,y=-h)$

Answer 1

Answer: (A), (B), (C)

Time taken to reach from $(-\ell ,-h)$ to $(\ell ,-h)$ is given by
$2\ell =\frac{1}{2}a{t}^2$
$20=\frac{1}{2}(10)t^2$
$t=2sec$
$\vec{L}=mvh\hat{k}=math\hat{k}$
$\vec{L}=(0.2)(10)2(1)\hat{k}=4\hat{k}$ [ when particle passes through $(\ell ,-h)]$
$\vec{\tau }=\frac{d\vec{L}}{dt}=mah\hat{k}=(0.2)(10)(1)\hat{k}=2\hat{k}$ (always)