chap. 10: rotational motion - texas a&m...

Chap. 10: Rotational

Motion

1

I. Rotational Kinematics II. Rotational Dynamics - Newton’s Law for

Rotation III. Angular Momentum Conservation (Chap.

10)

Rotational Motion

1. Remember how Newton’s Laws for translational

motion were studied:

1. Kinematics (x = x0 + v0 t + ½ a t2 )

2. Dynamics (F = m a)

3. Momentum Conservation

2. Now, we repeat them

again, but for rotational

motion:

1. Kinematics (q, w, a)

2. Dynamics (t = I a)

3. Angular Momentum

Part III

2

Rotational Motion

2

c m

2

2

1

2

1 wIMM g H v

Icm = Large v = Small

3

Rotational Motion

Rotational Motion

Rolling Motion w/o SlippingRolling Motion w/o Slipping

FasterFaster

Instantaneously restInstantaneously rest

Instantaneous axisInstantaneous axis

w

w

Newton’s 2nd Law of Motion

Translational Motion

Rotational Motion

)(integralnet tv

m

Fa

Quick Question: All shapes have the same mass,

but the speeds are different. Why?

Unlike a = F/m, even if F and m are given,

we have to calculate t and I.

4

Rotational Motion Rotational Motion

Mechanical Energy ConservationMechanical Energy Conservation

K = Km + KMK = Km + KM

Rotational Motion

ParallelParallel--axis Theoremaxis Theorem

Icm = ½ MR02

d

2 MdII cm

Idisk = Icylinder

Rotational Motion

t = r x F

q

F

r

tz

y

xF

r

z

Rotational Motion

Torque due to Gravity?Torque due to Gravity?

)(sin2

q

t

mg xl

F r

r

x

F

lCM

mass m

? F r

t

Rotational Motion

Vector Nature of Vector Nature of

Angular QuantitiesAngular QuantitiesKinematical variables to describe the rotational motion:

Angular position, velocity and acceleration

c.w. or c.c.w. rotation (like +x or –x direction in 1D)

Vector natures!

)(

)(

)(

2rad/s k

dt

dk

rad/s k dt

dk

rad R

l

ˆˆ

ˆˆ

wa

qw

q

x

y

z

R.-H. Rule

>0 or <0

Newton’s 2nd Law for Rigid-body Rotation

at

In et

Krot = (1/2) I w2

qtt s in Fr

6

Rotational Motion

9

Rotational Motion

10

Try This.

Rotational Motion

Angular Momentum

pprL

FFr

v m pIL

v

mI

t

w

w

11

Rotational Motion

Rotational Dynamics

dt

pd amF

dt

Ld I

at (torque )

m

I

a

ax

No external force

Momentum conservation

No external torque

L conservation

F

F

r

Speeding up

12

Rotational Motion

Which one has a bigger I?

Fp

Fp

13

Rotational Motion

Which one has a bigger I?

15

Rotational Motion

x

y

)()( 21 tF tF xx

fx ,fx ,x ,ix ,i p ppp 2121

Ia: big Ib: small z

17

Angular Momentum Conservation

The gravitational force and normal force don’t do anything about the rotation. The net torque is zero.

0)()( 21 tF tF xx

Rotational Motion

Ia: big Ib: small z

)(aw

)(bw

18


We observe the rotational speed increases. angular velocity (“omega”) increases. Why?

0ˆ )( Ng

kτττ i

x

y

)()( 21 tF tF xx


0)()( 21 tF tF xx

Rotational Motion

Ia: big Ib: small z

)(aw

)(bw

19


x

y

)()( 21 tF tF xx


0)()( 21 tF tF xx

0ˆ )( Ng

kτττ i

bbaa

ba

I I

L L

ww

Rotational Motion

2

cm

2

2

1

2

1 wIMMgH v

Icm = Large v = Small

bbaa

ba

I I

L L

ww

0ˆ )( Ng

kτττ i

Ia: big Ib: small

z

)(aw

)(bw

Angular momentum conservation Energy conservation

20

Conceptual Training

Rotational Motion

A person stands, hands at the side, on a platform that is

rotating at a rate of 1.60 rev/s. If the person now raises

his arms to a horizontal position, the speed of rotation

decreases to 0.800 rev/s.

(a) Why does the speed of

rotation decrease?

Explain using the two

key words;

external torque

angular momentum

(b) By what factor has the

moment of inertia of the

person changed?

(c) Compare Ki and Kf Iiwi Ifwf

21

Example 1

Rotational Motion

22

Example 2 P10-43 Simplified Model

Rotational Motion

23

Angular Momentum Conservation in Life

Rotational Motion

24

Angular Momentum Conservation?

Yes.

Let’s try to see in the next page.

E. 10.42

Rotational Motion

m2

m1

m1

wi

wf

Ii=MR2

If=MRf2

Rf

27

Iiwi Ifwf

Rotational Motion

A uniform disk turns at 7.00 rev/s around a

frictionless spindle. A non-rotating rod, of the

same mass (m) as the disk and length (l) equal

to the disk’s diameter, is dropped onto the disk.

They then both turnaround the spindle with

their centers superposed. There is no slipping

between the rod and the disk. What is the

moment of inertia of the disk+rod system about

the axis?

28

Pre-Example 2 E. 10.45

Rotational Motion

Example 2








angular velocity (in rev/s) of the disk+rod

system about the axis?

29

Iiwi Ifwf

E. 10.45

Rotational Motion








angular velocity (in rev/s) of the disk+rod

system about the axis?

Example 2

32

Iiwi Ifwf

Rotational Motion

Example 3

34

Example 10.10

Example 10.11

Example 10.12

P10-43

P10-45

P10-50

P10-88

P10-91

Rotational Motion

Example 3 Hint

35

Example 10.10

Example 10.11

Example 10.12

P10-43

P10-45

P10-50

P10-88

P10-91

Rotational Motion

Example 3 – Deep Impact Suppose a meteor (m = 7.0 x 1010 kg) struck the Earth at the

equator with a speed v = 1.0 x 104 m/s and remained stuck. By what

factor would this affect the rotational frequency of the Earth?

41

r

v

E a rthRr

m w(before)

prLm

Basic concept:

“By what factor..”

R = w(after)/w(before)

Lm =

LE = IEarth w(before)

Lm+E = Imeteor+Earth w(after)

REarth m v sinq

(into page)

(into page)

(out of page)

Emmm LLL

Imeteor+Earth = Imeteor + IEarth

Table 9.2

A spinning figure

skater pulls his arms

in as he rotates on the

ice. As he pulls his

arms in, what

happens to his

angular momentum L

and kinetic energy K?

A. L and K both increase.

B. L stays the same; K increases.

C. L increases; K stays the same.

D. L and K both stay the same.

Q10.11

The four forces shown all have the

same magnitude: F1 = F2 = F3 = F4.

Which force produces the greatest

torque about the point O (marked by

the blue dot)?

A. F1

B. F2

C. F3

D. F4

E. not enough information given to decide

Q10.1 F1

F2

F3

F4

O

Which of the four forces shown

here produces a torque about O that

is directed out of the plane of the

drawing?

A. F1

B. F2

C. F3

D. F4

E. more than one of these

Q10.2 F1

F2

F3

F4

O

A plumber pushes

straight down on the

end of a long wrench

as shown. What is the

magnitude of the

torque he applies about

the pipe at lower right?

A. (0.80 m)(900 N)sin 19°

B. (0.80 m)(900 N)cos 19°

C. (0.80 m)(900 N)tan 19°

D. none of the above

Q10.3

Q10.4

What is the torque that this force applies about the origin?

A. zero

B.

C.

D.

E.

ˆ ˆ24 18 N mi + j •

ˆ ˆ24 18 N mi j •

ˆ ˆ18 24 N mi + j •

ˆ ˆ18 24 N mi j •

A force ˆ ˆ4 3 NF = i j acts on an object at a point

ˆ6 m.r = klocated at the position

A. m2g = T2 = T1

B. m2g > T2 = T1

C. m2g > T2 > T1

D. m2g = T2 > T1

E. none of the above

Q10.5

A glider of mass m1 on a frictionless horizontal track is connected

to an object of mass m2 by a massless string. The glider accelerates

to the right, the object accelerates downward, and the string rotates

the pulley. What is the relationship among T1 (the tension in the

horizontal part of the string), T2 (the tension in the vertical part of

the string), and the weight m2g of the object?

A lightweight string is wrapped

several times around the rim of a

small hoop. If the free end of the

string is held in place and the hoop

is released from rest, the string

unwinds and the hoop descends.

How does the tension in the string

(T) compare to the weight of the

hoop (w)?

A. T = w

B. T > w

C. T < w

D. not enough information given to

decide

Q10.6

A solid bowling ball rolls

down a ramp.

Which of the following forces

exerts a torque on the bowling

ball about its center?

A. the weight of the ball

B. the normal force exerted by the ramp

C. the friction force exerted by the ramp

D. more than one of the above

E. The answer depends on whether the ball rolls without

slipping.

Q10.7

Rotational MotionAngular

Momentum

Challenge Change in


dt

Ld I

at (torque )

59


Momentum

Challenge: Change in

Angular Momentum (I)

x

y

z

R.-H. Rule

x

y

z

R.-H. Rule

wf Iwi Iwf wi

dt

Ld I

at (torque )

F

60


Momentum

Only the direction of L changes;

The magnitude of L does not change.

Out of page (+x axis)

F

iL

fL

if LLL

dt

Ld I

at (torque )

61

Challenge: Change in

Angular Momentum (II)


Momentum and Torque

dt

Ld I

at (torque)

62


Momentum and Torque

z

y

x

Spinning

L Spinning Axis

Rope

x

z

Rotation (in xy plane) of

Spinning Bicycle Wheel

63

Rotational Motion

Example 1

64

Rotational Motion

Recap: Example 1

Calculate the torque on the 2.00-m long

beam due to a 50.0 N force (top) about

(a) point C (= c.m.)

(b) point P


beam due to a 60.0 N force about


(b) point P


beam due to a 50.0 N force (bottom) about


(b) point P

qtt s in Fr

0ˆ )( ? ]W hy[ Ng

kτττ i

65

Rotational Motion

L = r p sinq Direction and magnitude of angular momentum?

Magnitude: L = r p sinq

Lm = r (mv) sinq = R0 m v

r

q v

66

Rotational Motion



Lm = r (mv) sinq = R0 m v Lm = ?

q

w

r

v

E a rthRr

r

q v

m

67

Rotational Motion



Lm = r (mv) sinq = R0 m v Lm = REarth m v sinq

q

w

r

v

E a rthRr

r

q v

m

68

Rotational Motion

(a) What is the angular momentum of a 3.00-kg uniform cylindrical

grinding wheel of radius 20 cm and height 40 cm when rotating

at 1500 rpm?

(b) How much torque is required to stop it in 10 s?

H 0.4H

w

if

if

tt

LL

td

Ld b

ILa

t

w

)(

)(Hints:

69

Example 2

chap. 10: rotational motion - texas a&m...

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