Moment of Inertia
13
Moment of Inertia Moment of Inertia
description
Moment of Inertia. Moment of Inertia. Units: kg m 2 The quantity ∑m i r i 2 , which is the proportionality constant between angular acceleration and net torque Torque Equation Depends on the axis of rotation. Torque— τ net Moment of Inertia—I Angular Acceleration— α - PowerPoint PPT Presentation
Transcript of Moment of Inertia
Moment of InertiaMoment of Inertia
Moment of InertiaMoment of Inertia
Units: kg m2
The quantity ∑miri2, which is the
proportionality constant between angular acceleration and net torque
Torque Equation
Depends on the axis of rotation
( )mrii i
2
Angular Versus LinearAngular Versus Linear
Torque—τnet
Moment of Inertia—I Angular Acceleration
—α Newton’s Second Law
—τnet=I α
Force—Fnet
Mass—m Acceleration—a Newton’s Second Law
—Fnet=ma
Moment of InertiaMoment of Inertia
Newton’s second law is easy to write, but we need to know the object’s moment of inertia.
Unlike mass, we cannot measure the moment of inertia. We must calculate it.
The moment of inertia is the sum over all the particles of the system.
Moment of InertiaMoment of Inertia
I r m I r dmii
m 2
02
Where r is the distance from the rotation axis.
I x y dm ( )2 2
If we let the axis of rotation be the z-axis then the moment of inertia becomes:
Problem 1Problem 1
Find the moment of inertia of a thin, uniform rod of length L and mass M that rotates about a pivot at one end.
Answer Problem 1Answer Problem 1
I x dm
dmMLdx
IML
x dx
IML
x
I ML
L
L
2
2
03
0
2
313
( )
( )
( )( )
Problem 2Problem 2
Find the moment of inertia of a circular disk of radius R and mass M that rotates on an axis passing through its center.
Answer Problem 2Answer Problem 2dm
MAdA
dmMR
rdrMRrdr
I r dm rMRrdr
IMR
r dr
IMR
rMR
disk
disk
R
disk
R
2 2
2 22
23
0
2
4
0
2
22
2
2
24
12
( )
( )
Parallel-Axis TheoremParallel-Axis Theorem
The moment of inertia depends on the axis of rotation.
If you need to know the moment of inertia for an axis that is not through the center of mass you can use an axis parallel to that and the parallel-axis theorem.
I I Mdcm 2
Where d is the distance from the axis to the axis through the center of mass.
Problem 3Problem 3
The engine in a small airplane is specified to have a torque of 60Nm. This engine drives a 2.0 m long, 40 kg propeller. On start-up, how long does it take the propeller to reach 200 rpm?
Answer Problem 3Answer Problem 3
The propeller can be considered a rod that rotates about its center.
The moment of inertia of a rod about its center is 1/12 ML2.
I = 1/12(40 kg)(2.0 m)2 = 13.33 kg m2
α = τ/I = 60Nm/13.33kgm2 = 4.50 rad/s2
trad srad s
20 9 04 5
4 62
. /. /
. sec
Problem 4Problem 4
A 10.0 cm diameter, 5.0 kg disk turns on an axle. A vertical cable attached to the edge of the disk exerts a 100 N force but, initially, a pin keeps the disk from rotating. What is the initial angular acceleration of the disk when the pin is removed?
mg
100N
Axle
