Question 1

The vector relation between linear velocity ($\vec{v}$), angular velocity ($\vec{\omega}$), and position vector ($\vec{r}$) for a rotating particle is:

Accepted Answer

The linear velocity is given by the cross product of angular velocity and position vector: $\vec{v} = \vec{\omega} 	imes \vec{r}$. The order of vector cross product is crucial.

Question 2

What is the rotational analogue of force in linear motion?

Accepted Answer

Torque (or moment of force) produces angular acceleration in rotational motion, just as force produces linear acceleration in translational motion.

Question 3

The moment of inertia of a uniform thin rod of mass $M$ and length $L$ about an axis passing through its center and perpendicular to its length is:

Accepted Answer

By integrating $dI = x^2 dm$ from $-L/2$ to $+L/2$, the moment of inertia about the central perpendicular axis is $I = \frac{1}{12}ML^2$.

Question 4

The SI unit of radius of gyration is:

Accepted Answer

Radius of gyration ($k$) is defined as the effective distance from the axis of rotation where the entire mass of the body can be assumed to be concentrated ($I = Mk^2$). Therefore, its SI unit is the unit of distance, which is metre ($m$).

Question 5

If no external torque acts on a rotating system, which of the following physical quantities remains constant?

Accepted Answer

According to the principle of conservation of angular momentum, when the net external torque acting on a system is zero, its total angular momentum remains constant ($	au{ext} = \frac{dL}{dt} = 0 \implies L = 	ext{constant}$).

System of Particles and Rotational Motion