Question

Which of the following statements are correct for the instantaneous axis of rotation? This question has multiple correct options.
A. Acceleration of every point lying on the axis must be equal to zero
B. Velocity of a point distance r from the axis is equal to \[r\omega \]
C. If the moment of inertia of a body about the axis is \[I\] and angular velocity is \[\omega \], then kinetic energy of the body is equal to \[I{\omega ^2}/2\]
D. Moment of inertia of a body is least about instantaneous axis of rotation among all the parallel axes

Answer 1

Hint: In case of rotational motion, acceleration is directed inside. To find the velocity of a point from the axis, find the linear velocity. A rotating body possesses both the translational motion and rotational motion.

Complete step by step answer:
Instantaneous rotating axis moves through A for a rolling wheel. When the speed of A is up vertically i.e.\[{a_{\text{A}}} \ne 0\] thus option A is wrong.
Speed of a point with distance r of the axis is the same as \[r\omega \] with regard to the axis, but at that moment every point on the instantaneous rotation axis rests. Velocity of that point is equal to \[r\omega \].
\[{I_o}\] be the MOI about point \[{\text{O}}\].
Moment of inertia for A,
\[I = {I_o} + m{r^2}\]
Thus, the instantaneous rotating axis moment of inertia is higher than probable minimal value.
So total energy:
\[K.E\, = \,K.{E_{{\text{Rotational}}}} + K.{E_{{\text{Translational}}}}\]
$K.{E_{\text{T}}} = \dfrac{1}{2}{I_o}{\omega ^2} + \dfrac{1}{2}m{v^2} \\
\implies K.{E_{\text{T}}} = \dfrac{1}{2}{I_o}{\omega ^2} + \dfrac{1}{2}m{r^2}{\omega ^2} \\
\implies K.{E_{\text{T}}} = \dfrac{1}{2}\left[ {{I_o} + m{r^2}} \right]{\omega ^2} \\
\therefore K.{E_{\text{T}}} = \dfrac{1}{2}I{\omega ^2} \\$

So, the correct answers are “Option B and C”.

Additional Information:
Inertia: Inertia, property of a body by which it opposes an entity which tries to move it, or if it moves, to alter the size or direction of its speed. Inertia is a passive property which allows a body to do little but oppose proactive agents such as strengths which torques. A moving body does not continue to travel because of its inertia, but because there is no force to slow it down, change or speed up its direction.
Moment of inertia: The moment of inertia of a rigid body is a quantity that defines the torque needed for an angular acceleration on the rotation axis. It is also known as the mass moment of inertness or an Angular mass inertia or rotational inertia; it similarly specifies the strength required for a desired acceleration. It depends on the mass distribution of the body and on the axis chosen, whereby the body rate of rotation takes more time.

Note:
In order to solve this problem, first remember that in rotational motion, a special kind of acceleration related to the rotating body which is called centripetal acceleration, is directed inside. Again, it is important to note that the velocity which corresponds to a point at a certain distance from the axis, is the linear velocity. Kinetic energy of a body is the sum of translational and rotational kinetic energy.