# Moment of Inertia - From Fourier Analysis to Fourier Transform

```FoS2 Physics Lecture 16
Moment of inertia
Torque
Rolling without slipping
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Kinetic energy for objects undergoing rotation
Kinetic energy of a collection of moving particles, if they are all rotating with the same angular velocity ω (“rigid body”):
(rigid body)
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Center of Mass and Moment of Inertia : case of continuous mass distribution
Example #1: a solid cone (Calc-MVC connection point)
a) Derive an expression for the mass of a uniform solid cone given in the
Figure below, in terms of R, h and density ρ.
b) Calculate the height ycm of the center of mass of a uniform solid cone given
in the Figure, in terms of height h.
c) Calculate the moment of inertia for rotation about y-axis, Iyy
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Some common objects (derivations in the textbook)
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Solid and hollow sphere – try these for yourself
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Rotating and translating body
The kinetic energy of a rotating and translating rigid body is
K = 1/2 Mvcm2 + 1/2 Icmω2.
=
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A challenge from the “Veritasium” channel
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Some predictions
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Experimental result
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How to make rotation: Rotational dynamics
Which of the three equal-magnitude forces
in the figure is most likely to loosen the
bolt?
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(Direction – right hand rule)
Three ways to calculate torque!
LC question
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Rotation considerations with Newton’s second law
Newton’s second law:
Careful ! True of MAGNITUDES only
Which Newton’s 2nd law do you need?
Both! 
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Rolling without slipping is an interesting case
TRANSLATION ONLY:
ROTATION ONLY:
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Rolling without slipping is an interesting case
TRANSLATION ONLY:
ROTATION ONLY:
PUT THEM TOGETHER:
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Rolling without slipping is an interesting case
TRANSLATION ONLY:
ROTATION ONLY:
PUT THEM TOGETHER:
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Rolling without slipping is an interesting case
Radius of the wheel
and by taking a time
derivative,
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```