Acceleration in Plane Motion:

If a body is performing plane motion and acceleration of the pole A and angular acceleration of the body is known, then we may calculate the acceleration of any other point B as illustrated in Figure.

We have

r_B  = r_A  + r_AB

∴ d ² r_B /dt ² = d ² r_A /dt ² + d ² r_AB /dt ²

or a _B  = a _A  + a _AB

where,

a _A  = Acceleration of pole A, and

a _AB  = Acceleration of B with respect to A.

Figure 5.38

But, we know that B rotates about A, with ω and α as angular velocity and acceleration respectively.

∴ a _AB  = a _AB t^ + a AB n^

=          AB . α       +           AB . ω²

Perpendicular AB            From B → A

∴ d ² r_B/ dt ² = d ² r_A/ dt ² + d/dt. (ω × r_AB )

= d ² r_A/ dt ² = (d ω¯/ dt)  r_AB + ω¯  (d r_AB /dt)

∴ a_B  = a _A  + α × r_AB  + ω r_AB

= a _A  + a AB t^ + a AB  n^

These components are shown in Figure and by combining these vectors we may get the final acceleration of point B.