1. Chapter 10: Rotational Motional About a Fixed Axis
Review of Angular Quantities & Motion and Torque (10-1,10-2,10-3, 10-5)
Solving Problems in Rotational Dynamics (10-6, 10-7)
Determining Moments of Inertia, Conservation of Angular Momentum, and Rotational Kinetic Energy (10-8,10-9, 10-10)
Rotational + Translational Motion (10-11, 10-12)
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2. Remember NO CLASS Monday!
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3. Which leads to the analog of Newton’s 2nd Law for angular motion:
Where we are...
Definition of torque
Which leads to the analog of Newton’s 2nd Law for angular motion:
The moment of inertia serves the role of mass
Today we’ll concentrate on “I” and explore angular momentums and rotational energy.
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4. Analytically Determining Moments of Inertia
The summation works very well for a discrete set of objects.
However if there is a continuous distribution throughout space we have to “go over to the limit of an infinite number of infinitesimal masses”, which is the integral.
Here dm represents the mass of an infinitesimal portion of the object at position R.
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5. Example 1: A Hollow Cylinder
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8. A Useful Theorem
The definition of I can be used to prove the parallel-axis theorem:
If I is the moment of inertia of a body of mass M about any axis, and
If ICM is the moment of inertia about a parallel axis passing through the center of mass but a distance of h away from the first axis then
I=ICM+Mh2
This is extremely useful, given ICM then I can quickly be calculated for any other parallel axis.
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9. A Second Useful Theorem
Also proven by the definition:
Useful for planar or two dimensional objects for which
The thickness can be ignored
The rotation axis is perpendicular to the surface
The perpendicular axis theorem:
The sum of the moments of inertia about any two perpendicular axes in the plane of the object equals the moment of inertia about a third axis which
passes through their intersection and
Is perpendicular to the plane of the object.
For the figure shown that is Iz=Ix+Iy
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10. Example 2: Ix for a Coin
The coin is a thin cylinder and has a moment of inertia Iz=(1/2)MRo2. What is the moment of inertia about Ix?
From the perp.-axis theorem we know that
Iz=(1/2)MRo2= Ix+ Iy.
But by symmetry Ix=Iy so
Iz=(1/2)MRo2= 2Ix 
Ix =(1/4)MRo2
And we can use the rotational second law about this new axis!
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11. Angular Momentum and Kinetic Energy
We’ve repeatedly seen the analogous behavior of linear and angular variables
xq, vw, aa and the eqs. of motion
This goes all the way to Newton’s 2nd Law:
SF = ma  St=Ia.
You shouldn’t be surprised to see analogs for momentum and energy!
P=mv  L=Iw (where L is defined as the angular momentum)
KE=(1/2)mv2  KE=(1/2)Iw2
We’ll spend the rest of this lesson exploring L or angular momentum and rotational kinetic energy.
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12. Angular Momentum and Its Conservation
The angular momentum for an object rotating about a fixed axis with moment of inertia I and angular velocity w is defined to be
L=Iw
the units for angular momentum are kg-m2/s
We can re-express the angular 2nd law with this variable assuming I does not change with time:
St=Ia=I(dw/dt)=d(Iw)/dt=dL/dt
If there is no external torque dL/dt is zero and L is a conserved quantity! Always a useful property!
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13. Conservation of Angular Momentum
In words: The total angular momentum of a rotating body remains constant if the net external torque acting on it is zero.
Since L=Iw this means that Iw=constant in the absence of a torque. Either the moment of inertia or the angular velocity may change but the product remains the same!
You can understand a ton of phenomena with this simple property of mechanics!
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16. Example 3: More Astronomy
Suppose a star, the same size (Ri=7x105km) as our sun, and revolving once every ten days collapsed to a neutron star of Rf=10km.
What would be the final rotation speed?
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17. The Vector nature of Angular Momentum
Actually it turns out (just like momentum angular momentum) angular momentum has direction.
For the simple case of rotation about a symmetry axis the direction of L is the same as w. In vector notation:
This is the case for a wheel or disk.
Remember if the angular velocity corresponds to
CCW motion it is said to positive and
CW is negative.
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18. Example 4: Walking on a Merry-go-round
Imagine a 60-kg person walking on the edge of a 6.0-m-diameter circular platform with a moment of inertia of 1800kgm2. The platform is initially at rest but the person starts running at 4.2m/s around the edge.
What is the angular velocity of the platform?
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21. Rotational Kinetic Energy
Ok we’ve seen the analog for equations of motion, the 2nd Law, and momentum
No surprise it also works for kinetic energy! In fact we can guess for a single particle.
The units work: kgm2/sec2=Joules
We can actually prove this quite easily.
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23. And the Work Energy Theorem.
For linear motion work done on an object changes the kinetic energy: W=KE2-KE1.
This is true for kinetic motion as well
This can be proven by starting with the line integral definition of work and applying it to a rotating rigid body.
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24. What is the angular velocity of the rod when it reaches the vertical?
Example 5: Rotating Rod
A rod of mass M pivots on a hinge as shown. The rod is initially at rest and then released.
What is the angular velocity of the rod when it reaches the vertical?
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