Resource Lesson
Thin Rods: Moment of Inertia
Printer Friendly Version
For linear, or translational, motion an object's resistance to a change in its state of motion is called its
inertia
and it is measured in terms of its mass, (kg). When a rigid, extended body is rotated, its resistance to a change in its state of rotation is called its
rotational inertia
, or
moment of inertia
. This resistance has a two-fold property.
the amount of mass present in the object and
the distribution of that mass about the chosen axis of rotation
Recall that the equation used to calculate the moment of inertia of a collection of discrete masses about an arbitrary axis of rotation is
where
r
is the perpendicular distance from the axis of rotation to each mass. We can use this same process for a continuous, uniform thin rod having a
mass per unit length (kg/m), λ
. To begin, let's divide our rod into sections having a constant mass, Δm
_{i}
, each a distance r
_{i}
from the pivot point.
Now let's make the Δm sections smaller and smaller, that is, let's take the limit as Δm → 0.
In general, for a continuous, rigid body, the moment of inertia is calculated with the equation
Unfortunately we cannot calculate the given integral because we can't integrate x
^{2}
with respect to "dm." We must either express
x in term of m
or
dm in terms of dx
. We will use the rod's uniform
mass per unit length (kg/m), λ
, to facilitate this substitution.
Now let's use this process to calculate the moment of inertia of a uniform, thin rod, rotated about its center of mass.
Below is a series of diagrams for a thin rod illustrating how the moment of inertia for the same object can change with the placement of the axis of rotation. Notice, that the farther the pivot point is from the object's center of mass, the greater its moment of inertia.
axis: far end of a thin rod
axis: one-fourth of the way from
the end of a thin rod
axis: center of a thin rod
These results would indicate that a thin rod would be most easily rotated about an axis through its center of mass (
I
= 4/48 mL
^{2}
= 1/12 mL
^{2}
) than about one of its far ends (
I
= 16/48 mL
^{2}
= 1/3 mL
^{2}
). Consider a majorette. If she twirls her baton about its center of gravity, for the same amount of torque she will achieve a greater rate of angular acceleration than if she twirls the same baton about a pivot closer to one of its ends. Consequently, a drum major maneuvering his mace or a member of the band's flag corps spinning their flag pole have to deal with larger moments of inertia and therefore have more difficulty accelerating their respective apparatus.
Parallel Axis Theorem
Using the
parallel axis theorem
,
we can calculate the rod’s moment of inertia about any point as long as we know that position's distance (
h
) from the object's center of mass. Let's practice by calculating the moment of inertia of a thin rod about its left end.
Notice that this is the same result that we would have obtained had we integrated our basic definition for the moment of inertia.
Related Documents
Lab:
Labs -
A Physical Pendulum, The Parallel Axis Theorem and A Bit of Calculus
Labs -
Calculation of "g" Using Two Types of Pendulums
Labs -
Conical Pendulums
Labs -
Conical Pendulums
Labs -
Conservation of Energy and Vertical Circles
Labs -
Conservation of Momentum in Two-Dimensions
Labs -
Density of an Unknown Fluid
Labs -
Introductory Simple Pendulums
Labs -
Kepler's 1st and 2nd Laws
Labs -
Loop-the-Loop
Labs -
Mass of a Paper Clip
Labs -
Moment of Inertia of a Bicycle Wheel
Labs -
Oscillating Springs
Labs -
Roller Coaster, Projectile Motion, and Energy
Labs -
Rotational Inertia
Labs -
Sand Springs
Labs -
Simple Pendulums: Class Data
Labs -
Simple Pendulums: LabPro Data
Labs -
Video LAB: A Gravitron
Labs -
Video LAB: Circular Motion
Labs -
Video LAB: Looping Rollercoaster
Labs -
Water Springs
Resource Lesson:
RL -
A Chart of Common Moments of Inertia
RL -
A Derivation of the Formulas for Centripetal Acceleration
RL -
A Further Look at Angular Momentum
RL -
Center of Mass
RL -
Centripetal Acceleration and Angular Motion
RL -
Conservation of Energy and Springs
RL -
Derivation of Bohr's Model for the Hydrogen Spectrum
RL -
Derivation: Period of a Simple Pendulum
RL -
Discrete Masses: Center of Mass and Moment of Inertia
RL -
Energy Conservation in Simple Pendulums
RL -
Gravitational Energy Wells
RL -
Hinged Board
RL -
Introduction to Angular Momentum
RL -
Kepler's Laws
RL -
LC Circuit
RL -
Magnetic Forces on Particles (Part II)
RL -
Period of a Pendulum
RL -
Rolling and Slipping
RL -
Rotary Motion
RL -
Rotational Dynamics: Pivoting Rods
RL -
Rotational Dynamics: Pulleys
RL -
Rotational Dynamics: Rolling Spheres/Cylinders
RL -
Rotational Equilibrium
RL -
Rotational Kinematics
RL -
Rotational Kinetic Energy
RL -
SHM Equations
RL -
Simple Harmonic Motion
RL -
Springs and Blocks
RL -
Symmetries in Physics
RL -
Tension Cases: Four Special Situations
RL -
The Law of Universal Gravitation
RL -
Thin Rods: Center of Mass
RL -
Torque: An Introduction
RL -
Uniform Circular Motion: Centripetal Forces
RL -
Universal Gravitation and Satellites
RL -
Vertical Circles and Non-Uniform Circular Motion
Review:
REV -
Review: Circular Motion and Universal Gravitation
Worksheet:
APP -
Big Al
APP -
Ring Around the Collar
APP -
The Baton Twirler
APP -
The Satellite
APP -
The See-Saw Scene
APP -
The Spring Phling
APP -
Timex
CP -
Center of Gravity
CP -
Centripetal Acceleration
CP -
Centripetal Force
CP -
Satellites: Circular and Elliptical
CP -
Torque Beams
CP -
Torque: Cams and Spools
NT -
Center of Gravity
NT -
Center of Gravity vs Torque
NT -
Circular Orbits
NT -
Falling Sticks
NT -
Pendulum
NT -
Rolling Cans
NT -
Rolling Spool
NT -
Rotating Disk
NT -
Spiral Tube
WS -
Basic Practice with Springs
WS -
Inertial Mass Lab Review Questions
WS -
Introduction to Springs
WS -
Kepler's Laws: Worksheet #1
WS -
Kepler's Laws: Worksheet #2
WS -
Moment Arms
WS -
Moments of Inertia and Angular Momentum
WS -
More Practice with SHM Equations
WS -
Pendulum Lab Review
WS -
Pendulum Lab Review
WS -
Practice: SHM Equations
WS -
Practice: Uniform Circular Motion
WS -
Practice: Vertical Circular Motion
WS -
Rotational Kinetic Energy
WS -
SHM Properties
WS -
Static Springs: The Basics
WS -
Torque: Rotational Equilibrium Problems
WS -
Universal Gravitation and Satellites
WS -
Vertical Circular Motion #1
TB -
Basic Torque Problems
TB -
Center of Mass (Discrete Collections)
TB -
Centripetal Acceleration
TB -
Centripetal Force
TB -
Moment of Inertia (Discrete Collections)
TB -
Rotational Kinematics
TB -
Rotational Kinematics #2
PhysicsLAB
Copyright © 1997-2017
Catharine H. Colwell
All rights reserved.
Application Programmer
Mark Acton