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

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

Moment of Inertia of Homogeneous Rigid Bodies with Different Geometries

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

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?

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

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

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

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

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?

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

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