Announcements1. Midterm 2 on Wednesday, Oct. 19.

2. Material: Chapters 7-11

3. Review on Tuesday (outside of class time)

4. I’ll post practice tests on Web

5. You are allowed a 3x5 inch cheat card

6. Go through practice exams & homework & class examples; understand concepts & demos

7. Time limit for test: 50 minutes

Conservation of energy (including rotational energy):

finalrotationalfinallinearfinitialrotationalinitiallineari

fi

KKUKKU

EE

,,,,

Again:

If there are no non-conservative forces: Energy is conserved.

Rotational kinetic energy must be included in energy considerations!

Connected cylinders.

Two masses m1 (5 kg) and m2 (10 kg) are hanging from a pulley of mass M (3 kg) and radius R (0.1 m), as shown. There is no slip between the rope and the pulleys.

(a) What will happen when the masses are released?

(b) Find the velocity of the masses after they have fallen a distance of 0.5 m.

(c) What is the angular velocity of the pulley at that moment?

Black board example 11.5

Torque

A force F is acting at an angle on a lever that is rotating around a pivot point. r is the ______________ between F and the pivot point.

This __________________ pair results in a torque on the lever

sin Fr

sinF

cosF

F

r

Black board example 11.6

Two mechanics are trying to open a rusty screw on a ship with a big ol’ wrench. One pulls at the end of the wrench (r = 1 m) with a force F = 500 N at an angle = 80 °; the other pulls at the middle of wrench with the same force and at an angle = 90 °.

What is the net torque the two mechanics are applying to the screw?

Particle of mass m rotating in a circle with radius r.

force Fr to keep particle on circular path.

force Ft accelerates particle along tangent.

Torque and angular acceleration Newton’s __________ law for rotation.

tt maF

Torque acting on particle is ________________ to angular acceleration : I

Work in rotational motion:

Definition of work:

Work in linear motion:

sFW

sdFdW

cos

sFsFW

sdFdW

Component of force F along displacement s. Angle between F and s.

___

___

W

dW

sdFdW

Torque and angular displacement .

Work and Energy in rotational motion

Remember work-kinetic energy theorem for linear motion:

22

2

1

2

1if mvmvW

There is an equivalent work-rotational kinetic energy theorem:

22 ___2

1___

2

1ifW

External work done on an object changes its __________ energy

External, rotational work done on an object changes its _______________energy

Linear motion with constant linear acceleration, a.

tavv xxixf

2

2

1tatvxx xxiif

)(222ifxxixf xxavv

tvvxx xfxiif )(2

1

Rotational motion with constant rotational acceleration,

_________f

_______________________f

___________________________2 f

________________f

Summary: Angular and linear quantities

_________RK

______

Kinetic Energy:

Torque:

Linear motion Rotational motion

2

2

1vmK

maF

Kinetic Energy:

Force:

Momentum: mvp __LAngular Momentum:

Work: sFW

_____WWork:

Rolling motion

Pure rolling:

There is no ___________

Linear speed of center of mass:

R

dt

dR

dt

dsvCM

Rolling motion

The _______ __________ of any point on the wheel is the same.

The linear speed of any point on the object changes as shown in the diagram!!

For one instant (bottom), point P has _______ linear speed.

For one instant (top), point P’ has a linear speed of ____________

Rolling motion of a particle on a wheel

(Superposition of ________ and ___________ motion)

Rolling

Rotation

Linear

+

=

Superposition principle:

Rolling motion = Pure _________ + Pure _______

Rolling motion

Kinetic energy

of rolling motion:____

2

1____

2

1CMIK

• Torque

• Angular momentum

• Angular momentum is conserved

Chapter 11: Angular Momentum part 1

Reading assignment: Chapter 11.4-11.6

Homework : (due Monday, Oct. 17, 2005):

Problems: 30, 41, 42, 44, 48, 53

Torque and the ______________

Thus far:

Torque sinFr

Torque is the _____________ between the force vector F and vector r

Fr

Torque and the vector product

Definition of vector product:

BAC

- The vector product of vectors A and B is the ___________.

- C is _________________ to A and B

- The __________________ of is C = A·B·sin

Torque and the vector product

BAC

Use the right hand rule to figure out the direction of C.

- __________ is C (or torque angular velocity , angular momentum L)

- _____________ finger is A (or radius r)

- ____________ finger is B (or force F)

Torque and the vector product

BAC

Rules for the vector product.

ABBA

If A is ______ to B then . Thus, 0BA

0AA

If A is _______ to B then BABA

CABACBA

)(

1.

2.

3.

4.

5. Magnitude of C = A·B·sin is equal to area of ______________ made by A and B

Torque and the vector product

BAC

Rules for the vector product (cont).

kBABAjBABAiBABABA xyyxxzzxyzzy

)()()( 6.

A force F = (2.00i + 3.00j) is applied to an object that is pivoted about a fixed axis aligned along the z-axis.

The force is applied at the point r = (4.00i + 5.00j).

Black board example 12.2

HW 21

(a) What is the torque exerted on the object?

(b) What is the magnitude and direction of the torque vector .

(c) What is the angle between the directions of F and r?