Question 475: Centre of Mass and Rotational Motion

Question

A particle of mass m is moving along the
side of a square of side ‘a’, with a uniform
speed v in the x-y plane as shown in the
figure :
Which of the following statements is
false
for the angular momentum $\overrightarrow L $ about the
origin ?
![](https://app-content.cdn.examgoal.net/fly/@width/image/1l7zd82l7/08aa0cff-95de-4203-bfe2-fe1213b63917/477db7a0-32ee-11ed-8cf6-c1445513adbd/file-1l7zd82l8.png?format=png)

Accepted Answer

The angular momentum is
$\overrightarrow L  = m(\overrightarrow r  	imes \overrightarrow v )$
For the particle moving from $D 	o A$, we have
$\overrightarrow L  = m\left[ {\left( {{R \over {\sqrt 2 }}\widehat i + {R \over {\sqrt 2 }}\widehat j} ight) 	imes v( - \widehat j)} ight]$
$ = {{mvR} \over {\sqrt 2 }}( - \widehat k)$
For the particle moving from $A 	o B$, we have
$\overrightarrow L  = m\left[ {\left( {{R \over {\sqrt 2 }}\widehat i + {R \over {\sqrt 2 }}\widehat j} ight) 	imes v\widehat i} ight]$
$ = {{mvR} \over {\sqrt 2 }}( - \widehat k)$
For the particle moving from $C 	o D$, we have
$\overrightarrow L  = mv 	imes $ ($\bot$ distance) $	imes$ $(\widehat k)$
$ = mv\left( {{R \over {\sqrt 2 }} + a} ight)(\widehat k)$
For the particle moving from $B 	o C$, we get
$\overrightarrow L  = mv 	imes $ ($\bot$ distance) $	imes$ $(\widehat k)$
$ = mv\left( {{R \over {\sqrt 2 }} + a} ight)(\widehat k)$
![](https://app-content.cdn.examgoal.net/fly/@width/image/1l3316jxk/061996f4-72f8-43b0-b9b3-cfaa54095619/3d6c8e80-d1f5-11ec-b83f-ebfea682138a/file-1l3316jxl.png?format=png)

Answer

$\overrightarrow L  = mv\left[ {{R \over {\sqrt 2 }} + a} \right]\widehat k$
when the
particle is moving from B to C.

Answer

$\overrightarrow L  = {{mv} \over {\sqrt 2 }}R\widehat k$
when the particle is moving from D to A.

Answer

$\overrightarrow L  =  - {{mv} \over {\sqrt 2 }}R\widehat k$
when the particle is
moving from A to B

Answer

$\overrightarrow L  = mv\left[ {{R \over {\sqrt 2 }} - a} \right]\widehat k$
when the
particle is moving from C to D.

Centre of Mass and Rotational Motion

A particle of mass m is moving along the side of a square of side ‘a’, with a uniform speed v in the x-y plane as shown in the figure : Which of the following statements is false for the angular momentum \overrightarrow L about the origin ?

Choose the correct answer: