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entire bar first translates by an amount dr so that A, the base point,
A
moves to its final position and point B moves to B', Fig. 2-23,b. The
bar is then rotated about A by an amount dθ so that B' undergoes a
relative displacement dr / BA and thus moves to its final position B. Due
to the rotation about A, dr / BA = r dθ and the displacement of B is
/ B A
dr B = dr A + dr / BA , 2-80

where dr - due to translation and rotation, dr - due to translation of
A
B
A, dr / BA - due to rotation about A.
55 Relative-Motion Analysis of Rigid Body: Velocity

To determine the relation between the velocities of points A and
B, it is necessary to take the time derivative of the position equation,
or simply divide the displacement equation by dt. This yields
dr B = dr A + dr / BA . 2-81
dt dt dt
The terms dr B / dt = v and dr A / dt = v are measured with
A
B
respect to the fixed x, y axes and represent the absolute velocities of
points A and B, respectively. Since the relative displacement is caused
by a rotation, the magnitude of the third term is
ɺ
dr / BA / dt = r dθ / dt = r θ = r ω where is the angular velocity of
/ BA
/ B A
/ BA
the body at the instant considered. We will denote this term as the
relative velocity v / BA , since it represents the velocity of B with respect
to A as measured by an observer fixed to the translating x', y' axes. In
other words, the bar appears to move as if it were rotating with an
angular velocity ω about the z' axis passing through A. Consequently,
v / B A has a magnitude of v / B A = ω r and a direction which is
/ B A
perpendicular to r / BA . Therefore we obtain the theorem about the
velocity of points of the rigid body:
v B = v A + v / B A . 2-82

The velocity of any point of a rigid body equals geometrical
sum the velocity of pole and the relative velocity this point around
the pole.

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