Physics 211 Lec 18: Center of Mass

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Physics 211 Lec 18 Center of Mass Moment of
Inertia

• Center of Mass (COM)
• The COM for a collection of matter is the average
  location of the mass.

Discrete form Group of particles each with their
own mass and coordinate location
Scalar Integral form take particles of
infinitesimally small mass dm with x,y,z coords.
Vector Integral form take particles of
infinitesimally small mass dm and position r
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Moment of Inertia (I)
Moment of inertia is a measure of the objects
resistance to rotation - mass resists linear
acceleration - moment of inertia resists
rotational acceleration
but
from the definition of density, so
Where does this equation for moment of inertia
come from?
Linear kinetic energy rotational kinetic energy
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Moment of Inertia Calculations - Shortcuts
Looking up results for common shapes see pg.
304
Using the Parallel Axis Theorem I ICM MD2
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Example Ch9 worked example 9.14Calculate the
center of mass for a straight rod of length L and
mass M if a) it has uniform mass per unit
length ?b) it has linearly increasing mass per
unit length (due to a variable diameter) ? ? x
where ? is a constant.
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Example Ch9 41See Figure P9.41. Find the
(x,y) coordinates for the center of mass of the
object.
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Example Ch10 24See Figure P10.24. Also,
model the tire as having 0.635cm uniformly thick
sidewalls and average 2.5 cm thick tread 20cm
wide. Assume a uniform density of 1.1x103 kg/m3.
Find the moment of inertia about the tires
central axis.
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Questions for Groups

• 9.20 Balance a meter stick on your index fingers
  at its ends. Bring your fingers together slowly.
  At what position on the meter stick do your
  fingers meet? Why?
• 9.26 (again) Does the center of mass of a rocket
  in free space accelerate? Explain.
• 10.12 and 13 Why does changing the axis of
  rotation of an object change its moment of
  inertia? What axis will give the minimum value
  for moment of inertia?