ENGR 215 ~ Dynamics Section 17.1

Moment of Inertia

• The translational aspects of motion are described by the equation:

• The rotational aspects of motion are described by the equation:

maF

IM

Golf Ball Drop Demo

Moment of Inertia

• Just as mass is a measure of a body’s resistance to acceleration inertia is a measure of a body’s resistance of a body to angular acceleration.

What is the purpose of the flywheel on this 1942 John Deere Model G tractor?

Definition of Moment of Inertia

• We define the moment of inertia as the integral of the “second moment” about an axis of all the elements of mass dm which compose a body?

• r = moment arm or perpendicular distance to the axis of rotation

dmrIm 2

Definition of Moment of Inertia

dmrIm 2

dVrIV

2

Differential Volume

dx dy dzdV • To obtain the moment of inertia we will only consider symmetrical bodies having surface which can be generated by revolving a curve about an axis.

Shell Elements

• If the shell having a height z, radius r, and a thickness dy is chosen then the volume is given by:

• We need z as a function of y to integrate.

dyzr dV 2

Disk Elements

• If the disk having a radius r and a thickness dz is chosen then the volume is given by:

• We need y as a function of z to integrate.

dzy dV 2

Lecture Example 1: Find the moment of inertia about the z- axis

Lecture Example 2: Find the moment of inertia about the y-axis

Parallel Axis Theorem

axes parallel between distance larperpendicu

body the of mass

G mass of center the through passing

plane z' the about inertia of moment

d

m

I

mdII

G

G2

Lecture Example 3: Find the Moment of Inertia for a slender rod length, L about its center of mass.

Lecture Example 4: Find the Moment of Inertia for a slender rod length, L about its end.

Lecture Example 5: Back to the golf ball drop!

Radius of Gyration

• Occasionally, the moment of inertia of a body about a specified axis is reported as the radius of gyration, k.

2mkI m

Ik

Lecture Example 6: The pendulum consists of two slender rods AB and OC which have a mass of 3 kg/m. The thin plate has a mass of 12 kg/m2. Determine the center of mass of the pendulum and the moment of inertia about an axis perpendicular to the page at Point G.