phy191 exam
Jocelyn yang 
Ch.12_1.pptxRotational Kinematics & Center of Mass
3/26/2108
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Get to Know Your Group Members
Introduce yourselves within your group
Suggested topics:
Name
Year in school
Why are you taking physics?
Major
Hometown
Goal for this semester
Favorite Pokemon
What do you hope to gain from your physics experience?
Would you rather fight 100 duck sized horses or 1 horse sized duck?
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Rotation of a Rigid Body
So far, motion has been treated rather thoroughly in this class
But that was only one type of motion: translational
Motion has many other forms, namely rotational, vibrational, and wave-like
The next few classes will be covering rotational motion
They recap everything we’ve done so far, but for spinning
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Good news, you’ve used nearly all of these concepts before
Bad news, all the letters are going to change, and they’ll mix together
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Nonuniform Circular Motion
For circular motion up to this point, we’ve considered accelerations only in the radial direction
These produce uniform circular motion
In nonuniform circular motion there is an acceleration in the tangential direction as well
The accelerations are then…
For constant at :
(maintains the circular motion)
(changes the speed)
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Consider Chapters 4.7 and 8.5 for more information
Divide regular kinematics by r to get circular
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Rigid Body Rotation
A rigid body is an extended object whose components don’t move relative to one another.
Can be a small group of particles at fixed positions, or a continuous body.
Pivot Point
Or Rotation axis
If the body is pivoted at some point, it is free to rotate about an axis through that point.
All points in the body have the same w, but different v.
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Rotational Kinematics
Rotational kinematics are essentially the same as circular motion, it’s just harder to see the circle.
The rotational kinematic equations are the same as the linear ones, just use the rotational counterparts:
Pivot
Point, P
Sign Convention:
CounterClockWise (CCW) is positive
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Whiteboard Problem 12-1
An ice skater holds their arms outstretched as they spin at 180 rpm.
What is the speed of their hands if they are 140cm apart? (LC)
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Whiteboard Problem 12-2
A high speed drill starts from rest and steadily increases its speed until it reaches 2000 rpm in a time of 0.50s.
What is the drill’s acceleration?
How many revolutions does the drill make in this time? (LC)
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Center of Mass
The center of mass is the point about which an unconstrained rigid body will rotate.
For a discrete collection of point masses…
The center of mass coordinates are:
N masses
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Since the distance from the origin is a vector, it is most useful to find the components of center of mass and then find the resulting distance.
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Whiteboard Problem 12-3
The masses shown below are connected by massless, rigid rods. Mass A is centered on the origin.
What is are the coordinates of the center of mass of this system? (LC)
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Good News about Continuous Bodies
To find the center of mass of a system, you consider the mass and location of each object.
What if you have a continuous object?
The center of mass coordinates are:
This type of problem can make for unpleasant integrals, but we won’t be doing (m?)any.
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Objects that are symmetric with uniform densities have the CoM at the geometric center.
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Vector Cross Product
larger
angle
Where:
What does the cross product mean in words?
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The Right Hand Rule
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The Right Hand Rule
Use your right hand.
No really.
With your four fingers extended and together, point them in the direction of the first vector of the cross product.
Rotate your hand until your palm is facing the second vector in the cross product.
Curls your fingers into a fist, moving through the angle θ to get to the second vector.
Your thumb will be the direction of the resultant vector.
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(for winners)
Whiteboard Problem 12-5
Evaluate the following cross products, giving direction as in to or out of the page.
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LC (mag only)
LC (mag only)
Where is 12-4?
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Vector Cross Product using a Determinant
Sometimes, you will be given the components of two vectors and be asked to find their cross product. This is done by taking a determinant; you may learn more about these in linear algebra but here’s the gist…
a, b, c, and d are just numbers
A 3x3 determinant can be expanded in to 3 2x2 determinants:
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Note, the negative sign; when you expand a determinant, the signs on the terms always alternate.
Consider the vectors:
This method directly gives you the components of the cross product vector. No angle, no RHR.
Vector Cross Product using a Determinant
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Whiteboard Problem 12-5 Cont.
Find the magnitude of the cross product between A and B. (LC)
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Whiteboard Problem 12-6
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