Sphere strikes hinged rod, e = 0.8

D'Alembert's principle converts a dynamics problem into a statics problem. A body in motion is in dynamic equilibrium if, in addition to the real forces and moments, we add inertia effects equal and opposite to the effective forces:

• Inertia force $-m\bar{\vec a}$ acting at the mass centre $G$.
• Inertia couple $-\bar I\vec\alpha$ (for plane motion).

For a rigid body in plane motion the equations of dynamic equilibrium are

$$\sum F_x - m\bar a_x = 0, \qquad \sum F_y - m\bar a_y = 0, \qquad \sum M_G - \bar I\alpha = 0$$

which are the same as $\sum \vec F = m\bar{\vec a}$ and $\sum M_G = \bar I\alpha$, but written as "sum of forces including inertia terms is zero". The moment equation can then be taken about any point, as in statics. Special cases: pure translation ($\alpha=0$): only the inertia force; rotation about a fixed axis through $G$: only the inertia couple $-I\alpha$.

   real forces + (-m a) at G + (-I alpha) couple  =>  equilibrium

Impact problem

Take moments about the hinge $A$: during the impact the hinge reaction passes through $A$ and the weights are non-impulsive, so the angular momentum about $A$ of sphere + rod is conserved. Let $L = 1.2$ m be the distance from $A$ to the point of impact (the lower end $B$), $v_s'$ the sphere velocity after impact (positive in its original direction) and $\omega$ the angular velocity of the rod.

Data

$m_s = 3$ kg, $v_s = 5$ m/s, $m_r = 8$ kg, $e = 0.8$

$$I_A = \frac13 m_rL^2 = \frac13(8)(1.2)^2 = 3.840\ \text{kg m}^2$$

Conservation of angular momentum about A

$$m_sv_sL = I_A\omega + m_sv_s'L \quad (1)$$

Restitution

Velocity of the rod's lower end is $\omega L$:

$$e = \frac{\omega L - v_s'}{v_s} \;\Rightarrow\; v_s' = \omega L - e\,v_s \quad (2)$$

Solving

Substituting (2) in (1):

$$m_sv_sL(1+e) = (I_A + m_sL^2)\,\omega$$ $$\omega = \frac{3(5)(1.2)(1+0.8)}{3.840 + 3(1.2)^2} = 3.97\ \text{rad/s}$$ $$v_s' = \omega L - e\,v_s = 3.97(1.2) - 0.8(5) = 0.76\ \text{m/s}$$

Answer: $\omega = 3.97$ rad/s; the sphere's velocity just after impact is $0.76$ m/s (in the original direction).