Kinetic Analysis | Sahithyan's Notes

D’Alembert’s Principle

By this principle, a particle or a system of particles moving with a constant acceleration is known as in dynamic equilibrium.

F - ma = 0

Here:

$ma$ is known as the inertia force which is fictious.

The equation can be rewritten as:

F = \sum m_ia_i = F = ma_G

Here:

Both translational and rotational aspects of the motion must be considered.

F=ma \;\;\; \text{and} \;\;\; M=I\alpha

Here:

\sum M_G = I_G \alpha

Here:

Let $O$ be a point other than $G$ .

\sum M_O = I_G \alpha + ma_G r_{OG}

T = \text{translational kinetic energy} + \text{rotational kinetic energy}

T = \frac{1}{2}mv_G^2 + \frac{1}{2}I_G\omega^2

The point on an object attached to a pivot where a perpendicular impact will produce no reactive shock at the pivot.

Translational and rotational motions cancel at the pivot when an impulsive blow is struck at the center of percussion.

l_{\text{P/G}} = \frac{k_\text{G}^2}{\text{OG}}

Here:

For a uniform rod:

l_{\text{P/G}} = \frac{L}{6}