Q.What is circular motion?

In rigid-body rotation every particle sweeps the same angle in the same time (same ω), but linear speed v = rω differs with the particle's distance r from the axis.

For a rigid body rotating about a fixed axis, all particles complete the same angular displacement in the same time interval, so every particle has the same angular velocity $\omega$. However, the linear (tangential) speed of a particle at distance $r$ from the axis is $v = r\omega$; since different particles lie at different $r$, they have different linear velocities.

UCM: constant speed on a circle; velocity direction keeps changing, giving a centripetal acceleration.

Uniform circular motion (UCM) is the motion of a particle along a circular path with constant speed. Although the speed $|v|$ is constant, the direction of velocity changes continuously (it is always tangential to the circle), so the particle has a non-zero acceleration directed towards the centre (centripetal acceleration $a = v^2/r$).

In uniform circular motion, speed is constant, so tangential acceleration is zero.

In uniform circular motion (U.C.M.), the speed of the particle stays constant even though its velocity direction keeps changing. Tangential acceleration $a_t = \dfrac{dv}{dt}$ measures the rate of change of speed, which is zero here — only a centripetal (radial) acceleration $a_c = v^2/r$ exists, directed towards the centre, responsible for changing the direction of motion, not its magnitude.

Angular displacement $\theta=\omega t$, with $\omega=v/r$ and $r=D/2$.

For a particle in uniform circular motion with tangential (linear) speed $v$ on a circle of diameter $D$ (radius $r=D/2$), the angular velocity is

$\omega=\frac{v}{r}=\frac{v}{D/2}=\frac{2v}{D}$

The total angular displacement in time $t$ is

$\theta=\omega t=\frac{2v}{D}\,t=\frac{2vt}{D}$

Checking the options: (A) $vt$ has units of length, not angle — wrong. (B) $\dfrac{v}{D}t$ misses the factor of 2 from $r=D/2$. (C) $\dfrac{vt}{2D}$ has an extra spurious factor of $1/4$ relative to the correct value. Only (D) matches the derivation.