Angular acceleration and linear acceleration I have a small confusion. I learned very recently that all particles of a rotating body have the same angular acceleration but different linear acceleration (same for velocity as well).
But how is this possible?
I find difficulty understanding this though I know that it is almost simple. Can you please help me understanding this?
 A: Consider a rod held vertically with a pivot at its base. If the rod is allowed to fall to one side each point on the rod turns through the same angle in a given amount of time. This is shown in the diagram bellow: 

To say otherwise would be against all experience. The definition  of angular acceleration is the rate of change of angular velocity or the second derivative of angle with respect to time i.e. $\frac{d^2\theta}{dt^2}$. But since for all points the angle they change is the same in a given amount of time their angular velocity must also be the same and hence the angular acceleration for each point on the rod is also the same.
Now to linear velocity. The distance travailed by any given point is given by $x=\theta r$  (this is simply arc length) where r is the distance of that point from the axis of rotation in this case the pivot. Linear velocity is the derivative of this with respect to time: $\frac{dx}{dt}=r\frac{d\theta}{dt}$. Since this depends on the distance of the point from the axis of rotation ($r$) which increases as you move along the rod away from the pivot this is not the same for each point on the rod. Similarly linear acceleration is given by $\frac{d^2x}{dt^2}=r\frac{d^2\theta}{dt^2}$ and again it to depends on distance from the axis of rotation and thus is not the same for every point on the rod.
