physics 1501: lecture 22, pg 1 physics 1501: lecture 22 today’s agenda l announcements çhw#8: due...

Physics 1501: Lecture 22, Pg 1

Physics 1501: Lecture 22Physics 1501: Lecture 22TodayToday’’s Agendas Agenda

AnnouncementsHW#8: due Oct. 28

Honors’ studentssee me Wednesday at 2:30 in P-114

TopicsRolling motionAngular MomentumFigure Skating Day !!! + other fun demos …


Torque, Work, Kinetic EnergyTorque, Work, Kinetic Energy We can define torque as: = r r x FF

= r F sin X = y FZ - z FY

Y = z FX - x FZ

Z = x FY - y FX

We find the work : W = Kinetic Energy of rotation: K = ½ I

Recall the Work Kinetic-Energy Theorem: K = WNET

So for an object that rotates about a fixed axis:

rr

FF

x

y

z


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

VCM


Rolling MotionRolling Motion Cylinders of different I rolling down an inclined plane:

h

v = 0

= 0

K = 0

R

K = - U = Mgh

v = R

M


Rolling...Rolling...

Use v= R and I = cMR2 .

So:

The rolling speed is always lower than in the case of simple

sliding since the kinetic energy is shared between CM motion

and rotation.

hoop: c=1

disk: c=1/2

sphere: c=2/5

etc...cc

cc


Lecture 22, Lecture 22, ACT 1ACT 1Rolling MotionRolling Motion

A race !!

Two cylinders are rolled down a ramp. They have the same radius but different masses, M1 > M2. Which wins the race to the bottom ?

A) Cylinder 1

B) Cylinder 2

C) It will be a tie

M1

h

M?

M2

Active Figure


Example :Example : Rolling MotionRolling Motion A cylinder is about to roll down an inclined plane. What is its

speed at the bottom of the plane ?

M

h

Mv ?

Ball has radius R

M M

M

M

M


Example :Example : Rolling MotionRolling Motion Use conservation of energy.

Ei = Ui + 0 = Mgh

Ef = 0 + Kf = 1/2 Mv2 + 1/2 I 2

= 1/2 Mv2 + 1/2 (1/2MR2)(v/R)2

Mgh = 1/2 Mv2 + 1/4 Mv2

v2 = 4/3 g h

v = ( 4/3 g h ) 1/2


Consider a roller coaster.

We can get the ball to go around the circle without leaving the loop.

Note:Radius of loop = RRadius of ball = r

Example :Example : Roller CoasterRoller Coaster


How high do we have to start the ball ?Use conservation of energy.Also, we must remember that the minimum

speed at the top is vtop = (gR)1/2

E1 = mgh + 0 + 0E2 = mg2R + 1/2 mv2 + 1/2 I2

= 2mgR + 1/2 m(gR) + 1/2 (2/5 mr2)(v/r)2

= 2mgR + 1/2 mgR + (2/10)m (gR) = 2.7 mgR

1

2


How high do we have to start the ball ?

E1 = mgh + 0 + 0E2 = 2.7 mgRmgh = 2.7 mgRh = 2.7 Rh = 1.35 D

(The rolling motion added an extra 2/10 R to the height: without it, h = 2.5 R)

1

2


Chap. 11: Angular MomentumChap. 11: Angular Momentum When we write = I we are really talking about

the z component of a more general vector equation. (we normally choose the z-axis to be the the rotation axis.)

z = Izz

We usually omit the

z subscript for simplicity.

z

z

z

Iz


Angular MomentumAngular Momentum

: angular momentum

r

mF

An object of mass m is rotating in a circular path under the action of a constant torque:


Angular MomentumAngular Momentum

The torque acting on an object is equal to the time rate of change of the object’s angular momentum

The angular momentum of an system is conserved when the net external torque acting on the system is zero. That is, when = 0, the initial angular momentum equals the final angular momentum.

Lf = Li


Angular Momentum:Angular Momentum:Definitions & DerivationsDefinitions & Derivations

We have shown that for a system of particles

Momentum is conserved if

What is the rotational version of this ??

The rotational analogue of force F F is torque

Define the rotational analogue of momentum pp to be

angular momentum

p=mv


Definitions & Derivations...Definitions & Derivations...

First consider the rate of change of LL:

So (so what...?)


Definitions & Derivations...Definitions & Derivations...

Recall that

Which finally gives us:

Analogue of !!

EXT


What does it mean?What does it mean?

where and

In the absence of external torquesIn the absence of external torques

Total angular momentum is conservedTotal angular momentum is conserved

Active torque Active angular momentum


i

j

Angular momentum of a rigid bodyAngular momentum of a rigid bodyabout a fixed axis:about a fixed axis:

Consider a rigid distribution of point particles rotating in the x-y plane around the z axis, as shown below. The total angular momentum around the origin is the sum of the angular momentum of each particle:

rr1

rr3

rr2

m2

m1

m3

vv2

vv1

vv3

We see that LL is in the z direction.

Using vi = ri , we get

L=I

(since ri , vi , are

perpendicular)

Analogue of p = mv !!


Angular momentum of a rigid bodyAngular momentum of a rigid bodyabout a fixed axis:about a fixed axis:

In general, for an object rotating about a fixed (z) axis we can write LZ = I

The direction of LZ is given by theright hand rule (same as ).

We will omit the ”Z” subscript for simplicity,and write L = I

z

LZ = I


Example: Two DisksExample: Two Disks

A disk of mass M and radius R rotates around the z axis with angular velocity 0. A second identical disk, initially not rotating, is dropped on top of the first. There is friction between the disks, and eventually they rotate together with angular velocity F.

0

z

F

z



First realize that there are no external torques acting on the two-disk system.Angular momentum will be conserved !

Initially, the total angular momentum is due only to the disk on the bottom:

0

z

2

1



First realize that there are no external torques acting on the two-disk system.Angular momentum will be conserved !

Finally, the total angular momentum is dueto both disks spinning:

F

z

21



Since LINI = LFIN

0

z

F

z

LINI LFIN

An inelastic collision,since E is not

conserved (friction) !

1/2 MR2 MR2 F

F = 1/2


Demonstration ofDemonstration ofConservation of Angular MomentumConservation of Angular Momentum Figure Skating :

A

z

B

z

Arm Arm

IA < IB

A > B

LA = LB


Angular Momentum ConservationAngular Momentum Conservation

A freely moving particle has a definite angular momentum about any given axis.

If no torques are acting on the particle, its angular momentum will be conserved.

In the example below, the direction of LL is along the z axis, and its magnitude is given by LZ = pd = mvd.

y

x

vv

d

m


Example: Bullet hitting stickExample: Bullet hitting stick

A uniform stick of mass M and length D is pivoted at the center. A bullet of mass m is shot through the stick at a point halfway between the pivot and the end. The initial speed of the bullet is v1, and the final speed is v2.

What is the angular speed F of the stick after the collision? (Ignore gravity)

v1 v2

M

F

before after

mD

D/4


Example: Bullet hitting stick...Example: Bullet hitting stick...

Conserve angular momentum around the pivot (z) axis! The total angular momentum before the collision is due

only to the bullet (since the stick is not rotating yet).

v1

D

M

before

D/4m



Conserve angular momentum around the pivot (z) axis! The total angular momentum after the collision has contributions

from both the bullet and the stick.

where I is the moment of inertia

of the stick about the pivot.

v2

F

after

D/4



Set LBEFORE = LAFTER using

v1 v2

M

F

before after

mD

D/4


Example: Throwing ball from stoolExample: Throwing ball from stool

A student sits on a stool which is free to rotate. The moment of inertia of the student plus the stool is I. She throws a heavy ball of mass M with speed v such that its velocity vector passes a distance d from the axis of rotation. What is the angular speed F of the student-stool

system after she throws the ball ?

top view: before after

d

vM

I I

F


Example: Throwing ball from stool...Example: Throwing ball from stool...

Conserve angular momentum (since there are no external torques acting on the student-stool system):LBEFORE = 0 , Lstool = IF

LAFTER = 0 = Lstool - Lball , Lball = Iballball = Md2 (v/d) = M d v

0 = IF - M d v

top view: before after

d

vM

I I

F

F = M v d / I

physics 1501: lecture 22, pg 1 physics 1501: lecture 22 today’s agenda l announcements çhw#8: due...

Documents