PHY 113 A Fall 2012 -- Lecture 19 110/15/2012

PHY 113 A General Physics I9-9:50 AM MWF Olin 101

Plan for Lecture 19:

Chapter 11 – angular momentum

1. Vector cross product

2. Angular momentum of a rotating rigid object

3. Conservation of angular momentum

PHY 113 A Fall 2012 -- Lecture 19 210/15/2012

PHY 113 A Fall 2012 -- Lecture 19 310/15/2012

How to make objects rotate.

Define torque:

t = r x F

t = rF sin q

r

F

q

αarτFraF

Imm

qsinr

q

F sin q

Previously:

PHY 113 A Fall 2012 -- Lecture 19 410/15/2012

PHY 113 A Fall 2012 -- Lecture 19 510/15/2012

Vector cross product; right hand rule

qsinBACBAC

ˆ ˆ ˆ ˆ ˆ ˆ 0ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ

i i j j k ki j j i kj k k j ik i i k j

PHY 113 A Fall 2012 -- Lecture 19 610/15/2012

ˆ ˆ ˆ ˆ ˆ ˆ 0ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ

i i j j k ki j j i kj k k j ik i i k j

For unit vectors:

i

k

j

PHY 113 A Fall 2012 -- Lecture 19 710/15/2012

y z x yx zx y z

y z x yx zx y z

A A A AA AA A A

B B B BB BB B B

ˆ ˆ ˆˆ ˆ ˆ

i j kA B i j k

More details of vector cross products:

y z z y x z z x x y y xA B A B A B A B A B A Bˆ ˆ ˆ A B i j k

PHY 113 A Fall 2012 -- Lecture 19 810/15/2012

iclicker exercise:What is the point of vector products

A. To terrify physics studentsB. To exercise your right handC. To define an axial vectorD. To keep track of the direction of rotation

PHY 113 A Fall 2012 -- Lecture 19 910/15/2012

prvrvrarτFr

aF

dtd

dtmd

dtdmm

m

From Newton’s second law:

prvr dtd

dtmd

ConsiderIs this

A. Wrong?B. Approximately right?C. Exactly right?

iclicker exercise:

PHY 113 A Fall 2012 -- Lecture 19 1010/15/2012

From Newton’s second law – continued – conservation of angular momentum:

(constant)

0 0 If

:Define

L

Lτ

prL

prτFr

dtd

dtd

PHY 113 A Fall 2012 -- Lecture 19 1110/15/2012

Torque and angular momentum

Define angular momentum:

For composite object: L = Iw

dtd

dtdItotal

Lωτ

prL

Newton’s law for torque:

In the absence of a net torque on a system,

angular momentum is conserved.

PHY 113 A Fall 2012 -- Lecture 19 1210/15/2012

A student sits on a rotatable stool holding a spinning bicycle wheel with angular momentum Li. What happens when the wheel is inverted?

iclicker exercise:

(a) The student will remain at rest.

(b) The student will rotate counterclockwise.

(c) The student will rotate clockwise.

counterclockwise

PHY 113 A Fall 2012 -- Lecture 19 1310/15/2012

More details:

wheelbf

wheelwheelbf

wheelibiwheelfbf

LL

LLL

LLLL

2

0

PHY 113 A Fall 2012 -- Lecture 19 1410/15/2012

Other examples of conservation of angular momentum

mm

d1 d1

mm

d2 d2

I1=2md12 I2=2md2

2

I1w1=I2w2 w2=w1 I1/I2

w1 w2

PHY 113 A Fall 2012 -- Lecture 19 1510/15/2012

What about centripetal acceleration?

w

ar = v2/R = w2 R

PHY 113 A Fall 2012 -- Lecture 19 1610/15/2012

Webassign problem:A disk with moment of inertia I1 is initially rotating at angular velocity wi. A second disk having angular momentum I2, initially is not rotating, but suddenly drops and sticks to the second disk. Assuming angular moment to be conserved, what would be the final angular velocity wf?

21

1

211

III

III

if

fi

ww

ww