1. -Calculation of Moments of Inertia for Rigid Objects of Different Geometries -Parallel Axis Theorem
AP Physics C
Mrs. Coyle

2. Remember:
Rotational kinetic energy of the rigid object is the sum of the energies of all its particles:

3. Moment of Inertia of a Rigid Object

4. NOTE
For objects of certain geometric shapes, (ie. cylinders, hoops, rods, spheres), we will use geometric relationships to evaluate the integral.
Note that for the following examples that the axis of rotation will coincide with the axis of symmetry of the objects.

5. Moment of Inertia of a Uniform Thin Hoop – for Axis perpendicular to the plane at the Center of the Hoop
R is constant

6. Densities are used to evaluate integrals for objects of uniform density
Volumetric Mass Density –> mass per unit volume: r = m / V
Face Mass Density –> mass per unit area of a sheet of uniform thickness, t : s = m/A= rt
Linear Mass Density –> mass per unit length of a rod of uniform cross-sectional area: l = m / L = rA

7. Moment of Inertia of a Uniform Rigid Rod for Axis passing through center of mass
= m / L = dm/dx
dm = l dx
Let r be x

8. Moment of Inertia of a Uniform Solid Cylinder for axis along z axis
For concentric shells with radius r, thickness dr and length L

13. Parallel-Axis Theorem
To find I , if the axis of rotation does not coincide with the axis of symmetry but is parallel to the axis through the center
of mass of the object.

14. Parallel Axis Theorem
I = ICM + MD 2 ICM , moment of inertia about the axis through the center of mass
D , distance from the center of mass axis to the arbitrary axis

15. Moment of Inertia for a Rod Rotating Around One End

17. Total Moment of Inertia of an Object
For a uniform density object made up of various shapes the total moment of inertia is the sum of the moments of inertia for the individual objects.

18. Example #25
A uniform thin solid door has height 2.20m, width 0.870m and mass 23.0kg. Find its moment of inertia fro rotation on one hinge across its bottom. Is any piece of data unnecessary? (Hint: Assume that all the mass is concentrated across the bottom of the door along the same level as the hinge)
Ans: 5.80 kg m2 , h is unnecessary data.

19. Example #23

20. Example #23
Three identical thin rods, each of length L and mass m, are welded perpendicular to one another as shown. The assembly is rotated about an axis that passes through the end of one rod and is parallel to another. Determine the moment of inertia of this structure.
Ans: (11/12)mL2

21. Example # 27
The density of the Earth at any distance r from its center, is approximately: ρ= [14.2 – 11.6 (r/R)] x 10 3 km/m3 where R is the radius of the Earth. Show that this density leads to a moment of inertia I= MR2 about an axis through the center, where M is the mass of the Earth. Hint: start with thin spherical shell equation: dI= (2/3) r2 dm