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Lecture 16. Chapter 12 Extend the particle model to rigid-bodies U nderstand the equilibrium of an extended object. Analyze rolling motion Understand rotation about a fixed axis. Employ “conservation of angular momentum” concept. Goals:. Assignment: HW7 due March 25 th

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Lecture 16
• Chapter 12
• Extend the particle model to rigid-bodies
• Understand the equilibrium of an extended object.
• Analyze rolling motion
• Understand rotation about a fixed axis.
• Employ “conservation of angular momentum” concept

Goals:

Assignment:

• HW7 due March 25th
• After Spring Break Tuesday:

Catch up

Rotational Dynamics: A child’s toy, a physics playground or a student’s nightmare
• A merry-go-round is spinning and we run and jump on it. What does it do?

What principles would apply?

• We are standing on the rim and our “friends” spin it faster. What happens to us?
• We are standing on the rim a walk towards the center. Does anything change?

Rotational Variables
• Rotation about a fixed axis:
• Consider a disk rotating aboutan axis through its center:
• How do we describe the motion:

(Analogous to the linear case )

v = w R

x

R

Rotational Variables...
• Recall: At a point a distance R away from the axis of rotation, the tangential motion:
• x =  R
• v = R
• a =  R
Comparison to 1-D kinematics

Angular Linear

And for a point at a distance R from the rotation axis:

x = R v =  R aT =  R

Here aT refers to tangential acceleration

System of Particles (Distributed Mass):
• Until now, we have considered the behavior of very simple systems (one or two masses).
• But real objects have distributed mass !
• For example, consider a simple rotating disk and 2 equal mass m plugs at distances r and 2r.
• Compare the velocities and kinetic energies at these two points.

w

1

2

1 K= ½ m v2 = ½ m (w r)2

w

2 K= ½ m (2v)2 = ½ m (w 2r)2

System of Particles (Distributed Mass):
• Twice the radius, four times the kinetic energy
• The rotation axis matters too!

+

+

m2

m1

m1

m2

• If an object is not held then it will rotate about the center of mass.
• Center of mass: Where the system is balanced !
• Building a mobile is an exercise in finding

centers of mass.

mobile

System of Particles: Center of Mass
• How do we describe the “position” of a system made up of many parts ?
• Define the Center of Mass (average position):
• For a collection of N individual point like particles whose masses and positions we know:

RCM

m2

m1

r2

r1

y

x

(In this case, N = 2)

RCM = (12,6)

(12,12)

2m

m

m

(0,0)

(24,0)

Sample calculation:
• Consider the following mass distribution:

XCM = (m x 0 + 2m x 12 + m x 24 )/4m meters

YCM = (m x 0 + 2m x 12 + m x 0 )/4m meters

XCM= 12 meters

YCM= 6 meters

System of Particles: Center of Mass
• For a continuous solid, convert sums to an integral.

dm

r

y

where dmis an infinitesimal

mass element.

x

VCM

Connection with motion...
• So for a rigid object which rotates about its center of mass and whose CM is moving:

For a point p rotating:

Rotation & Kinetic Energy
• Consider the simple rotating system shown below. (Assume the masses are attached to the rotation axis by massless rigid rods).
• The kinetic energy of this system will be the sum of the kinetic energy of each piece:
• K = ½m1v12 + ½m2v22 + ½m3v32 + ½m4v42

m4

m1

r1

r4

r2

m3

r3

m2

m4

m1

r1

r4

m3

r2

r3

m2

Rotation & Kinetic Energy
• Notice that v1 = w r1 , v2 = w r2 , v3 = w r3 , v4 = w r4
• So we can rewrite the summation:
• We recognize the quantity, moment of inertia orI, and write:
Calculating Moment of Inertia

where r is the distance from the mass to the axis of rotation.

Example:Calculate the moment of inertia of four point masses

(m) on the corners of a square whose sides have length L,

about a perpendicular axis through the center of the square:

m

m

L

m

m

Calculating Moment of Inertia...
• For a single object, Idepends on the rotation axis!
• Example:I1 = 4 m R2 = 4 m (21/2 L / 2)2

I1= 2mL2

I2= mL2

I= 2mL2

m

m

L

m

m

dr

r

R

L

dm

r

Moments of Inertia
• For a continuous solid object we have to add up the mr2contribution for every infinitesimal mass element dm.
• An integral is required to find I:

Solid disk or cylinder of mass M and radius R, about perpendicular axis through its center.

I = ½ M R2

• Some examples of I for solid objects:

Use the table…

Ball 1

Ball 2

Exercise Rotational Kinetic Energy
• ¼
• ½
• 1
• 2
• 4
• We have two balls of the same mass. Ball 1 is attached to a 0.1 m long rope. It spins around at 2 revolutions per second. Ball 2 is on a 0.2 m long rope. It spins around at 2 revolutions per second.
• What is the ratio of the kinetic energy

of Ball 2 to that of Ball 1 ?

Ball 1

Ball 2

Exercise Rotational Kinetic Energy
• K2/K1 = ½ m wr22 / ½ m wr12 = 0.22 / 0.12 = 4
• What is the ratio of the kinetic energy of Ball 2 to that of Ball 1 ?

(A) 1/4 (B) 1/2 (C) 1 (D) 2 (E) 4

Exercise Work & Energy
• Strings are wrapped around the circumference of two solid disks and pulled with identical forces, F, for the same linear distance, d. Disk 1 has a bigger radius, but both are identical material (i.e. their density r = M / V is the same). Both disks rotate freely around axes though their centers, and start at rest.
• Which disk has the biggest angular velocity after the drop?

W =F d = ½ I w2

(A)Disk 1

(B)Disk 2

(C)Same

w2

w1

F

F

start

d

finish

Exercise Work & Energy
• Strings are wrapped around the circumference of two solid disks and pulled with identical forces for the same linear distance. Disk 1 has a bigger radius, but both are identical material (i.e. their density r = M/V is the same). Both disks rotate freely around axes though their centers, and start at rest.
• Which disk has the biggest angular velocity after the drop?

W =F d = ½ I1w12 = ½ I2w22

w1 = (I2 / I1)½w2 and I2 < I1

(A)Disk 1

(B) Disk 2

(C)Same

w2

w1

F

F

start

d

finish

Lecture 16

Assignment:

• HW7 due March 25th
• For the next Tuesday:

Catch up

Lecture 16

Assignment:

• HW7 due March 25th
• After Spring Break Tuesday: Catch up