Scalar Moment of Inertia of A Simple Pendulum
Moment of inertia can be obtained by considering the movement of a mass at the end of a lightweight rod forming a simple pendulum, which can be studied using Newton's second law of motion. The weight of the mass is a force that accelerates it around the pivot point.
This weight also generates a torque T on the pendulum around the pivot point and the acceleration of the mass a = rα is defined by the angular acceleration α of the pendulum, therefore
where r is the length of the pendulum. The quantity I = mr2 is the moment of inertia of the pendulum mass around the pivot point.
In the same way, the kinetic energy of the pendulum mass is defined by its velocity v = rω using the angular velocity ω of the pendulum to yield
The angular momentum of the pendulum mass is given by
This shows that the quantity I = mr2 plays the same role for rotational movement, as mass does for translational movement. The moment of inertia of an arbitrarily shaped body is the sum of the values mr2 for all of the elements of mass in the body.
Read more about this topic: Inertia Tensor
Famous quotes containing the words moment, inertia, simple and/or pendulum:
“After I have said what is required by my vanity and my morality, I may find a moment for Truth.”
—Mason Cooley (b. 1927)
“What is wrong with priests and popes is that instead of being apostles and saints, they are nothing but empirics who say “I know” instead of “I am learning,” and pray for credulity and inertia as wise men pray for scepticism and activity.”
—George Bernard Shaw (1856–1950)
“Meantime the education of the general mind never stops. The reveries of the true and simple are prophetic. What the tender poetic youth dreams, and prays, and paints today, but shuns the ridicule of saying aloud, shall presently be the resolutions of public bodies, then shall be carried as grievance and bill of rights through conflict and war, and then shall be triumphant law and establishment for a hundred years, until it gives place, in turn, to new prayers and pictures.”
—Ralph Waldo Emerson (1803–1882)
“The pendulum oscillates between these two terms: Suffering—that opens a window on the real and is the main condition of the artistic experience, and Boredom ... that must be considered as the most tolerable because the most durable of human evils.”
—Samuel Beckett (1906–1989)