Question 74: Centre of Mass and Rotational Motion

Question

The position of an object having mass 0.1 kg as a function of time $t$ is given as
$\vec{r} = \left( 10 t^2 \hat{i} + 5 t^3 \hat{j} ight)$ m. At $t = 1$ s, which of the following statements are
correct
?
A.
The linear momentum $\vec{p} = \left( 2 \hat{i} + 1.5 \hat{j} ight)$ kg·m/s.
B.
The force acting on the object $\vec{F} = \left( 2 \hat{i} + 3 \hat{j} ight)$ N.
C.
The angular momentum of the object about its origin $\vec{L} = 15 \hat{k}$ J·s.
D.
The torque acting on the object about its origin $\vec{	au} = 20 \hat{k}$ N·m.
Choose the
correct
answer from the options given below:

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

A, B and C only

Answer

B, C and D only

Answer

A, C and D only

Answer

A, B and D only

Centre of Mass and Rotational Motion

Choose the correct answer: