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.

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

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

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.

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)

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

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,

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.

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

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

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.