Rotational Inertia

1
Rotational Inertia
2
Circular Motion

• Objects in circular motion have kinetic energy.
• K ½ m v2
• The velocity can be converted to angular
  quantities.
• K ½ m (r w)2
• K ½ (m r2) w2

m
r
w
3
Integrated Mass

• The kinetic energy is due to the kinetic energy
  of the individual pieces.
• The form is similar to linear kinetic energy.
• KCM ½ m v2
• Krot ½ I w2
• The term I is the moment of inertia of a
  particle.

4
Moment of Inertia Defined

• The moment of inertia measures the resistance to
  a change in rotation.
• Mass measures resistance to change in velocity
• 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.

5
Two Spheres

• A spun baton has a moment of inertia due to each
  separate mass.
• I mr2 mr2 2mr2
• If it spins around one end, only the far mass
  counts.
• I m(2r)2 4mr2

m
m
r
6
Mass at a Radius

• Extended objects can be treated as a sum of small
  masses.
• A straight rod (M) is a set of identical masses
  Dm.

• The total moment of inertia is
• Each mass element contributes
• The sum becomes an integral

distance r to rDr
length L
axis
7
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

8
Point and Ring

• The point mass, ring and hollow cylinder all have
  the same moment of inertia.
• I MR2
• All the mass is equally far away from the axis.

• The rod and rectangular plate also have the same
  moment of inertia.
• I (1/3) MR2
• The distribution of mass from the axis is the
  same.

M
R
R
M
M
M
length R
length R
axis
9
Parallel Axis Theorem

• Some objects dont rotate about the axis at the
  center of mass.
• The moment of inertia depends on the distance
  between axes.

• The moment of inertia for a rod about its center
  of mass

h R/2
M
axis
10
Spinning Energy

• How much energy is stored in the spinning earth?

• The earth spins about its axis.
• The moment of inertia for a sphere I 2/5 M R2
• The kinetic energy for the earth Krot 1/5 M R2
  w2
• With values K 2.56 x 1029 J

The energy is equivalent to about 10,000 times
the solar energy received in one year.
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