Which of the following statements are correct for instantaneous axis of rotation?

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 `romega`
C. if 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 `Iomega^(2)//2`
D. Moment of inertia of a body is least about nstantaneous axis of rotation among all the parallel axes.

Answer

B, C, D

Explanation

In case of a wheel rolling on a horizontal plane, instantaneous axis of rotation of the wheel passes through `A` (point of contact of wheel with plane). But at `A`, acceleration is vertically upwards. It means, acceleration at `A` has non-zero value. Hence, option (a) is incorrect.
If the point is at a distance `r` from the instantaneous axis of rotation, then its velocity (relative to the axis) will be equal to `romega`. But by definition, every point lying on the instantaneous axis of rotation is at rest at that instant. Therefore, the resultant velocity of the particle becomes equal to that relative velocity `romega`. Hence, option (b) is correct.
If distance of centre of mass of a body from the instantaneous axis of rotation is equal to `r` and if its moment of inertia parallel to instantaneous axis of rotation but passing through centre of mass is equal to `I_(0)` then `I=(I_(0)+mr^(2))`
Rotational translational `KE` of the body will be equal to `I_(0)omega^(2)//2` and translation `KE` of the body will be `mv^(2)//2` where `v =romega.` Therefore translational `KE` becomes equal to `mr^(2)omega^(2)//2`. Hence, the total `KE` becomes equal to `(I_(0)omega^(2)//2)+(mr^(2)omega^(2)//2)o=Iomega^(2)//2`.
Hence option c is correct.
In case of a wheel of the wheel from through `A` while the centre of rotation of mass is at distance `R` from `A`. In that case, the moment of instantaneous axis of rotation is equal to `I=(I_(0)+mR^(2))`, where `I_(0)` is momentum of inertia about the centroidal axis. Hence moment of inertia of the wheel instantaneous axis of rotation is greater than the minimum possible value. Hence, option (d) is incorrect.