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Two-Dimensional Rotational Kinematics
8.01 W09D1
Today’s Reading Assignment:
MIT 8.01 Course Notes: Chapter 16 Two Dimensional Rotational Kinematics
Sections 16.1-16.4 Chapter 17 Two Dimensional Rotational Dynamics
Sections 17.2
Announcements
No Problem Set due Week 9 No Math Review Week 9
Exam 2 Regrade Policy
If you would like any problem regraded on your exam, please write a note on the cover sheet and submit the exam no later then the following class. Exams have been photocopied so under no circumstances should you change any of your original answers. Any altered exams will automatically be graded zero.
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Rigid Bodies
A rigid body is an extended object in which the distance between any two points in the object is constant in time.
Springs or human bodies are non-rigid bodies.
Demonstrations
Baton: Center of Mass & Rotational Motion
Bicycle Wheel: Fixed Axis Rotation
Two-Dimensional Rotational Motion
Fixed Axis Rotation: Disc is rotating about axis passing through the center of the disc and is perpendicular to the plane of the disc.
Motion Where the Axis Translates: For straight line motion, bicycle wheel rotates about fixed direction and center of mass is translating
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Demo: Rotation and Translation of Rigid Body
Demonstration: Motion of a thrown baton
Translational motion: external force of gravity acts on center of mass
Rotational Motion: object rotates about center of mass
Demonstration: Bicycle Wheel: Fixed Axis Rotation
Recall: Translational Motion of the Center of Mass
Total momentum of system of particles
External force and acceleration of center of mass
psys = msys
Vcm
Fext
total =dpsys
dt= msys
dVcm
dt= msys
Acm
Main Idea: Rotational Motion about Center of Mass
Torque produces angular acceleration about center of mass
is the moment of inertia about the center of mass is the angular acceleration about center of mass Analagous to Newton’s Second Law for Linear Motion
Icm
τcm = Icm
αcm
αcm
Fext = macm
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Recall: Rotational Kinematics for Fixed Axis Rotation
A point like particle undergoing circular motion at a non-constant speed has (1) an angular velocity vector (2) an angular acceleration vector
Fixed Axis Rotation: Angular Velocity
Angular velocity Vector: Component Magnitude Direction
ω ≡ ω z k̂ ≡
dθdt
k̂
ω ≡ ω z ≡
dθdt
ω z ≡ dθ / dt > 0, direction + k̂
ω z ≡ dθ / dt < 0, direction − k̂
ω z ≡
dθdt
Concept Question: Angular Speed Object A sits at the outer edge (rim) of a merry-go-round, and
object B sits halfway between the rim and the axis of rotation. The merry-go-round makes a complete revolution once every thirty seconds. The magnitude of the angular velocity of Object B is
1. half the magnitude of the angular velocity of Object A . 2. the same as the magnitude of the angular velocity of Object A . 3. twice the the magnitude of the angular velocity of Object A . 4. impossible to determine.
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Fixed Axis Rotation: Angular Acceleration
Angular acceleration: Vector: Component: Magnitude: Direction:
α ≡ α z k̂ ≡
d 2θdt2 k̂
α z ≡
d 2θdt2 ≡
dω z
dt
α ≡ α z ≡
dω z
dt
α z ≡
dω z
dt> 0, direction + k̂
α z ≡
dω z
dt< 0, direction − k̂
Rotational Kinematics: Integral Relations
The angular quantities are exactly analogous to the quantities for one-dimensional motion, and obey the same type of integral relations
x, vx , and ax
θ ,ω z , and α z
ω z (t)−ω z (t0 ) = α z ( ′t )d ′t
′t =t0
′t =t
∫ ,
θ(t)−θ(t0 ) = ω z ( ′t )d ′t
′t =t0
′t =t
∫ .
Table Problem: Rotational Kinematics
a) Determine an expression for the z-component of the angular velocity, , for the interval [0, t1]. What is the value of ? b) Determine an expression for the angle for the interval [0, t1]. What is the value of the ? c) What is the value of the ?
A uniform disk of mass m and radius R. Define a coordinate system as shown in the figure below where the coordinates of a point P on the rim of the turntable are . The turntable is turned on at t = 0 when . An instant after time t1, the motor is turned off for the time interval [t1, t2]. The z-component of the angular acceleration of the turntable is given by the function
(R,θ(t))
θ(t = 0) = 0
α z (t) =
Bt ; 0 ≤ t ≤ t10 ; t1 < t ≤ t2
⎧⎨⎪
⎩⎪
ω z (t) ω z (t = t1)
θ(t)
θ(t = t2 )
θ(t = t1)
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Rigid Body Kinematics for Fixed Axis Rotation
Kinetic Energy and Moment of Inertia
Rigid Body Kinematics for Fixed Axis Rotation
Body rotates with angular velocity and angular acceleration
ω
α
Divide Body into Small Elements
Body rotates with angular velocity,
angular acceleration Individual elements of mass Radius of orbit Tangential velocity Tangential acceleration
Δmi
vθ ,i = riω z
aθ ,i = riα z
ri
ω ≡ ω z k̂
α ≡ α z k̂
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Rotational Kinetic Energy and Moment of Inertia
Rotational kinetic energy about axis passing through point S
Rotational Kinetic Energy:
Krot = Krot,i
i∑ = 1
2Δmiri
2
i∑⎛⎝⎜
⎞⎠⎟ω 2 = 1
2ISω
2
Krot,i =
12Δmivi
2 = 12Δmi(ri
2 )ω 2
IS ≡ Δmiri
2
i∑
Moment of Inertia about axis passing through point S :
Moment of Inertia: Continuous Body
Moment of Inertia about S : SI Unit: Continuous body: Moment of Inertia about S :
IS = Δmi(ri )
2
i=1
i=N
∑2kg m⎡ ⎤⋅⎣ ⎦
IS = dm (rdm )2
body∫
Δmi → dm, ri → rdm , →
body∫
i=1
i=N
∑
Table Problem: Moment of Inertia of Rod
Consider a thin uniform rod of length L and mass M. Calculate the moment of inertia about an axis that passes perpendicular to the rod through its center of mass of the rod,
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Discussion: Moment of Inertia
How does moment of inertia for fixed axis rotation about a symmetry axis of a symmetric body compare to the mass and the center of mass?
Total mass: scalar quantity Center of Mass: vector quantity Moment of Inertia about axis passing through S:
IS = dm (rdm )2
body∫
mtotal = dm
body∫
cm totalbody
1 dmm
= ∫R r
Concept Q.: Moment of Inertia Same Masses
All of the objects below have the same mass. Which of the objects"has the largest moment of inertia about the axis shown?"
(1) Hollow Cylinder (2) Solid Cylinder (3)Thin-walled Hollow Cylinder
Demo: Moment of Inertia of Various Objects
Moment of inertia of various cylindrical objects with the same mass but different mass distributions. Roll objects down incline plane. Note that this is both translational and fixed axis rotation which we will return to shortly.
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Concept Q.: Moment of Inertia about Center
Which has the smallest I about its center?
1) Ring (M,R)
2) Disk (M,R)
3) Sphere (M,R)
4) All have the same I
Concept Q.: Moment of Inertia about Different Axes
Which axis gives the largest I for the disk?
1)
2)
3)
4) All have the same I
Parallel Axis Theorem
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Parallel Axis Theorem
• Rigid body of mass m. • Moment of inertia about
axis through center of mass of the body.
• Moment of inertia about parallel axis through point S in body.
• dS,cm perpendicular distance between two parallel axes.
IS = Icm + mdS ,cm
2
Icm
IS
Concept Question: Moment of Inertia of Rod
Consider a thin uniform rod of length L and mass M. The moment of inertia about the center of mass is (1/12) ML2. Calculate the moment of inertia about an axis that passes perpendicular to the rod through one end of the rod is
a) Iend = (1/12) ML2. b) Iend = ML2. c) Iend = (1/4) ML2. d) None of the above.
Concept Question: Kinetic Energy A disk with mass M and radius R is spinning with angular speed ω about an axis that passes through the rim of the disk perpendicular to its plane. Moment of inertia about cm is (1/2)M R2, (you will calculate this in a homework problem). Its kinetic energy is
4. (1/4)M Rω2
5. (1/2)M Rω2
6. (1/4)M Rω
1. (1/4)M R2 ω2
2. (1/2)M R2 ω2
3. (3/4)M R2 ω2
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Table Problem: Pulley System and Energy
Using energy techniques, calculate the speed of block 2 as a function of distance that it moves down the inclined plane using energy techniques. Let IP denote the moment of inertia of the pulley about its center of mass. Assume there are no energy losses due to friction and that the rope does not slip around the pulley.