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Moment of Inertia

The moment of inertia measures an object's resistance to changes in rotation around an axis. It depends on how mass is distributed relative to the axis. For a single point mass, the moment of inertia is I = mr^2. The total moment of inertia of an object is the sum of its individual point-masses' moments of inertia. Common equations allow calculating the moment of inertia for basic shapes like rods, disks, spheres, and others, by integrating over all small mass elements. The parallel axis theorem and perpendicular axis theorem relate moments of inertia between different axes of rotation.

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700 views10 pages

Moment of Inertia

The moment of inertia measures an object's resistance to changes in rotation around an axis. It depends on how mass is distributed relative to the axis. For a single point mass, the moment of inertia is I = mr^2. The total moment of inertia of an object is the sum of its individual point-masses' moments of inertia. Common equations allow calculating the moment of inertia for basic shapes like rods, disks, spheres, and others, by integrating over all small mass elements. The parallel axis theorem and perpendicular axis theorem relate moments of inertia between different axes of rotation.

Uploaded by

senthilcae
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Moment of Inertia

Moment of Inertia Defined


 The moment of inertia measures the resistance to a
change in rotation.
• Change in rotation from torque
• Moment of inertia I = mr2 for a single mass

 The total moment of inertia is due to the sum of


masses at a distance from the axis of rotation.
N
I   mi ri 2
i 1
Two Spheres
 A spun baton has a moment
of inertia due to each
 separate mass.

• I = mr2 + mr2 = 2mr2


m m
r
 If it spins around one end,
only the far mass counts.

• I = m(2r)2 = 4mr2
Mass at a Radius
 Extended objects can be  The total moment of inertia is
treated as a sum of small
I   (Dm)r
2
masses.
 Each mass element
 A straight rod (M) is a set of contributes
identical masses Dm. Dm  ( M / L)Dr
I  ( M / L) r 2 Dr
distance r to r+Dr
 The sum becomes an
integral
length L L
I  ( M / L)  r 2 dr
0

axis I  ( M / L)( L3 / 3)  (1 / 3) ML2


Rigid Body Rotation
 The moments of inertia for many shapes can found
by integration.

• Ring or hollow cylinder: I = MR2


• Solid cylinder: I = (1/2) MR2

• Hollow sphere: I = (2/3) MR2


• Solid sphere: I = (2/5) MR2
Point and Ring
 The point mass, ring and  The rod and rectangular
hollow cylinder all have the plate also have the same
same moment of inertia. moment of inertia.

• I = MR2 • I = (1/3) MR2

 All the mass is equally far  The distribution of mass from


away from the axis. the axis is the same.

M
R M R M
M length R length R
axis
Playground Ride
 A child of 180 N sits at the  Assume the merry-go-round
edge of a merry-go-round is a disk.
with radius 2.0 m and mass • Id = (1/2)Mr2 = 320 kg m2
160 kg.  Treat the child as a point
 What is the moment of mass.
inertia, including the child? • W = mg, m = W/g = 18 kg.
• Ic = mr2 = 72 kg m2

m  The total moment of inertia is


M the sum.
r
• I = Id + Ic = 390 kg m2
Parallel Axis Theorem
 Some objects don’t rotate  The moment of inertia for a
about the axis at the center rod about its center of mass:
of mass.
h = R/2
M
 The moment of inertia
depends on the distance
axis
between axes.
(1 / 3) MR 2  I CM  M ( R / 2) 2
I  I CM  Mh 2
I CM  (1 / 3) MR 2  (1 / 4) MR 2
I CM  (1 / 12) MR 2
Perpendicular Axis Theorem
 For flat objects the rotational
Iy = (1/12) Ma2
moment of inertia of the axes
in the plane is related to the
moment of inertia
perpendicular to the plane.
Ix = (1/12) Mb2 b

M
a
Iz  Ix  Iy

Iz = (1/12) M(a2 + b2)


Spinning Coin
 What is the moment of  The moment of inertia of a
inertia of a coin of mass M spinning disk perpendicular
and radius R spinning on to the plane is known.
one edge? • Id = (1/2) MR2

 The disk has two equal axes


in the plane.
R  The perpendicular axis
M M theorem links these.
R
• Id = Ie + Ie = (1/2) MR2
• Ie = (1/4) MR2
Id Ie

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