1. Aim: How do we calculate the moment of inertia of a rigid body with a continuous mass distribution?

2. Finding the Moment of Inertia using calculus
I = ∫r2dm Moment of Inertia for an extended continuous object (rigid body)

3. Moment of Inertia of Homogeneous Rigid Bodies with Different Geometries

4. Linear, Area, and Volume Density
Linear Density: λ=dm/dl so dm = λdl Area Density: б=dm/dA so dm = бdA Volume Density: ρ=dm/dV sp dm = ρdV

5. Finding the moment of inertia about the center of mass axis of a Uniform Rod
I =∫r2dm
Express the linear mass density.
Write an expression for dm
What are the upper and lower limits of integration?

7. Finding the moment of inertia about the center of mass axis of a Uniform Solid Cylinder
I =∫r2dm
Are we dealing with linear mass density, area mass density, or volume mass density?
Express the area of a circle.
Express the volume of a cylinder.
Write an expression for dm
Select our lower and upper limits of integration

9. Parallel Axis Theorem
The Parallel Axis Theorem tells us how to calculate the moment of inertia of a rigid body about any axis parallel to the axis through the center of mass. It says I = Icm + Mh2 I=moment of inertia Icm=moment of inertia about an axis through the center of mass. h=distance between the two axes M=total mass of the object

10. Example 1-Using the Parallel Axis Theorem
Find the rotational inertia of a uniform rod about an axis through the end of the rod.
(1/3)ML2

12. Example 2-Using the Parallel Axis Theorem
Determine the moment of inertia of a uniform solid sphere of mass M and radius R about an axis that is tangent to the surface of the sphere. (The rotational inertia of a solid sphere about the center of mass is 2/5 MR2)
7/5MR2

14. Rotating Rod Problem
A uniform rod of length L and mass M is free to rotate on a frictionless pin through one end. The rod is released from rest in a horizontal position.
What is the angular speed at its lowest position?
What is the tangential speed of the lowest point on the rod?
What is the tangential speed of the center of mass?

15. Rotating Rod Problem
ω=√(3g/l) v = 1/2√3gl v = √3gl

16. Explain from an energy perspective
What type of energy does the rod have initially? Gravitational Potential Energy What type of energy does the rod have in the end? Rotational Kinetic Energy