Question 80: Centre of Mass and Rotational Motion

Question

When the position vector $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ changes sign as $-\vec{r}$, which one of the following vector will not flip under sign change?

Accepted Answer

Vectors are categorized into two types based on their symmetry :
1. Polar Vectors : These represent actual physical displacements. They flip their sign (direction) when the coordinate system is inverted ( $\vec{A} ightarrow-\vec{A}$ ).
2. Axial Vectors (Pseudo-vectors) : These usually represent rotations or are the result of a cross product of two polar vectors. They do not flip their sign under coordinate inversion $(\overrightarrow{\mathrm{B}} ightarrow \overrightarrow{\mathrm{B}})$.
Velocity is the rate of change of position: $\vec{v}=\frac{d \vec{r}}{d t}$
Since $\vec{r}$ changes to $-\vec{r}$, then $\frac{d(-\vec{r})}{d t}=-\vec{v}$.
So, when position vector changes sign then velocity vector also get flipped.
Momentum is defined as $\overrightarrow{\mathrm{p}}=\mathrm{m} \overrightarrow{\mathrm{v}}$.
Since $\overrightarrow{\mathrm{v}}$ flips to $-\overrightarrow{\mathrm{v}}$, then $\overrightarrow{\mathrm{p}}$ becomes $\mathrm{m}(-\overrightarrow{\mathrm{v}})=-\overrightarrow{\mathrm{p}}$.
So the linear momentum also flips if position vector changes sign.
Acceleration is the rate of change of velocity: $\vec{a}=\frac{d \vec{v}}{d t}$.
Since $\overrightarrow{\mathrm{v}}$ flips to $-\overrightarrow{\mathrm{v}}$, then $\frac{\mathrm{d}(-\overrightarrow{\mathrm{v}})}{\mathrm{dt}}=-\overrightarrow{\mathrm{a}}$.
So, the acceleration also gets flipped when position vector changes sign.
Angular momentum is defined by the cross product :
$\overrightarrow{\mathrm{L}}=\overrightarrow{\mathrm{r}} 	imes \overrightarrow{\mathrm{p}}$.
If we apply the sign change :
$$ \overrightarrow{\mathrm{L}^{\prime}}=(-\overrightarrow{\mathrm{r}}) 	imes(-\overrightarrow{\mathrm{p}}) $$
$$\Rightarrow $$ $$ \overrightarrow{\mathrm{L}^{\prime}}=(\overrightarrow{\mathrm{r}}) 	imes(\overrightarrow{\mathrm{p}}) $$
So, it does not flip. It is an axial vector. Hence, the correct option is (D).

Answer

Velocity

Answer

Linear momentum

Answer

Acceleration

Answer

Angular momentum

Centre of Mass and Rotational Motion

When the position vector \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} changes sign as -\vec{r}, which one of the following vector will not flip under sign change?

Choose the correct answer: