ENGR 215 ~ Dynamics Section 17.1

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MOMENT OF INERTIA Today’s Objectives: Students will be able to:

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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:

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

Definition of Moment of Inertia

Differential Volume 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.

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.

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

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.

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.