Question

Which of the following statement is correct for instantaneous axis of rotation?

• 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 moment of inertia of the body about the axis be I and angular velocity be $\omega$ then total kinetic energy of all body is equal to $\frac{1}{2}I{\omega }^2$
• D. Moment of inertia of a body is least about instantaneous axis of rotation among all the parallel axis

Answer 1

Answer: (B), (C)

The correct options are
B Velocity of a point distance $r$ from the axis is equal to $r\omega$
C If moment of inertia of the body about the axis be I and angular velocity be $\omega$ then total kinetic energy of all body is equal to $\frac{1}{2}I{\omega }^2$

The acceleration has component directed towards the center at all times during circular motion. Thus option A is wrong.
If a point is a distance r from the instantaneous axis, then its velocity relative to axis will be equal to $r\omega$. But by definition, every point on the instantaneous axis is at rest. Thus resultant velocity of particle becomes equal to that of relative velocity $r\omega$. Thus option B is correct.
We have moment of inertia I through instantaneous axis as $I=I_{cm}+m{r}^2$. Rotational and translational kinetic energy of the body are $\frac{1}{2}{I}_{cm}{\omega }^2$ and $\frac{1}{2}m{v}^2$. Here $v=r\omega$, thus we get translational kinetic energy as $\frac{1}{2}m{r}^2{\omega }^2$. Thus total kinetic energy is calculated as $\frac{1}{2}I{\omega }^2$ Thus option C is correct.
Moment of inertia is least about the centroidal axis. Thus option D is wrong.