Difference between a point mass and a rotating disk In Kinematics, we deal with particles (point masses) which do not have any rotation about their mass centres. In books, it is often mentioned that bodies are analysed as particles but only motion of an entire number of particles making a unit is considered. How does this statement valid for mechanical vibrations where we considered SDOF and MDOF systems? As an example, if we take the case of a rotating disk on a shaft, the equation of motion is $J\ddot{\theta}+K \theta=0$, then is it as I written above a system in which an entire number of particles is forming a unit (the disk)?
Here is an extract from the book "Vector mechanics for engineers: Dynamics, ninth edition by Beer"":

 A: When can I use the classical particle description?
Think of the particle description as the most basic model you can make. We neglect any features of the body and track the motion of its center of mass, giving us $3$ degrees of translational freedom. 
When does the particle description fail?
Generally speaking, if the features of the body are important to its dynamics, then the particle description is going to be lacking. 
Consider a fixed coordinate system in space, whose origin is initially at the center of mass of the object. Now set up another set of axes that point along three fixed directions along the body and whose origin is defined to be at the center of mass. If the relative angles between these two sets of axes - that of the spatial and bodily axes- ever change, then the body is rotating in space. We then cannot describe it as a particle having $3$ degrees of freedom. 
By neglecting any deformation of the macroscopic shape, the constituent particles of the body preserve their position in the body-centered frame. We call this rigid body dynamics. This model characterizes the motion of the macroscopic body through the translation and rotation of its body axes with respect to the spatially fixed frame, meaning we need up to $6$ degrees of freedom.
What about the rotating disc?
For a rotating disc, we require rigid body dynamics. Even if a flywheel's axis is off-center, its bodily coordinates are still rotating with respect to some spatial coordinates (see comment). By constraining the translational motion and confining the rotation along a single axis, we end up considering dynamics with respect to one angle, say $\theta$.
Further reading
For further information, I recommend David Tong's lecture notes on The Dynamics of Rigid Bodies, freely accessible on his website.
