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 ?

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) $$	imes$$ $$(\widehat k)$$
$$ = mv\left( {{R \over {\sqrt 2 }} + a} ight)(\widehat k)$$

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: