How to interpete the formula: $I_{G}\alpha = \dot{\vec{H}}$?

Question

I am having hard time to fully undestand what those formula's actually mean. I know the meaning of each element:

• I : moment of inertia
• H : angular momentum
• $\alpha$: angular acceleration
• $\omega$: angular velocity

$I_{G}\alpha = \dot{\vec{H}}$

$I_{G} \dot{\vec{\omega}}= \dot{\vec{H}}$

Could someone explain the meaning and/or give and example/context to clarify things?

Answer 1

Perhaps it is helpful to compare the equation to its linear counterpart.
$$ \begin{align} \vec{F}&=\dot{\vec{p}}=m\vec{a}=m\dot{\vec{v}}\\ \vec{\tau}&=\dot{\vec{H}}=I\vec{\alpha}=I\dot{\vec{\omega}} \end{align} $$ This linear formula says that the force on an object is equal the the rate of change of the linear momentum which is equal to the mass times the acceleration which is equal to the mass times the rate of change of the velocity.

In rotational dynamics, torque takes the place of force, the moment of inertia takes the place of mass, angular momentum takes the place of linear momentum, and angular velocity takes the place of linear velocity.

Answer 2

============Short answer===================

If you are familiar and understand Newton's second law:

$$F = m\cdot a = m\cdot \dot{v}$$

Then you should know that one way to interpret it, is that:

the acceleration $a$ of a body with mass $m$ is proportional to the sum of forces acting on the body.

Similarly, keeping in mind that it can be proven that $\dot{H} = M$, the equation you describe can be written is a similar fashion:

$$ M = I_G \cdot \alpha= I_G \cdot \dot{\omega}$$

Where:

• $M$ is the resultant moments acting on the body
• $I_G$ the moment of inertia
• $H$ the angular momentum
• $\omega$ the angular velocity [rad/s]
• $\alpha$ the angular acceleration $[rad/s^2]$

Essentially, what this equation says is that :

the angular acceleration $\alpha$ of a body with moment of inertia $I_G$ is proportional to the sum of moments acting on the body.

Observe that there is a direct correspondence between:

• forces F and moments M
• mass $m$ and mass moment of inertia $I_G$
• acceleration $a$ and angular acceleration $\alpha$

===================Long Answer==========================

I won't bother with the long answer, because it would be impossible to give the generic case (see rotating systems of reference) within the space of this answer.

You might want to read Dynamics textbooks (Meriam and Kraige , Beer etc)