derivation of moment of inertia of circle proof

Because r is the distance to the axis of rotation from each piece of mass that makes up the object, the moment of inertia for any object depends on the chosen axis.

We defined the moment of inertia I of an object to be I = ∑ i m i r i 2 I = ∑ i m i r i 2 for all the point masses that make up the object. This section is very useful for seeing how to apply a general equation to complex objects (a skill that is critical for more advanced physics and engineering courses). In this section, we show how to calculate the moment of inertia for several standard types of objects, as well as how to use known moments of inertia to find the moment of inertia for a shifted axis or for a compound object. In the preceding section, we defined the moment of inertia but did not show how to calculate it. Calculate the moment of inertia for compound objects.Apply the parallel axis theorem to find the moment of inertia about any axis parallel to one already known.Calculate the moment of inertia for uniformly shaped, rigid bodies.Note: The lower limit is -h because the left side of the rod is -h units away from the axis of rotation.By the end of this section, you will be able to: Substituting dI, (write the appropriate limits) Using the equation for dm, we substitute it into the first equation. Since the rod is uniform, the mass varies linearly with distance. Hence, we have to force a dx into the equation for moment of inertia. Now, lets find an expression for dm. Hey, there is a dm in the equation! Recall that we’re using x to sum. Now, we show our formula for the calculation for moment of inertia first: In other problem, the variable can be theta or r or … In this problem, we are summing from left of the axis to right of axis. Each of these slices have a mass of dm (remember your calculus lessons?) and length of dx.Imagining the rod is cut into infinitesimally many pieces of infinitesimally thin slices.GUIDE: (I’ll be guiding you through this as it may be the first time that you are seeing this) Derivation of moment of inertia of an uniform rigid rodĬalculate/derive the moment of inertia of an uniform rigid rod of length L and mass M about an axis perpendicular to the rod and passing through O, at an arbitrary distance h from one end.

Derivation of moment of inertia of circle proof