Chap. 9: Rotational Motionpeople.physics.tamu.edu/kamon/teaching/phys218/slide/... · 2013-04-01 · Rotational Motion Problem 3: (25 points) A grinding wheel turns at a varying angular

Chap. 9: Rotational

Motion

1

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

Rotation III. Angular Momentum Conservation (Chap.

10)

Rotational Motion

Rotational Motion I. Rotational Kinematics

II. Rotational Dynamics - Netwton’s

Law for Rotation

III.Angular Momentum Conservation

2

Rotational Motion

1. Remember how Newton’s Laws for

translational motion were studied:

1. Kinematics (x = x0 + v0 t + ½ a t 2 )

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

OUTLOOK

3

Rotational Motion

Can we explain this more

systematically and mathematically?

Fp

Fp

4

Rotational Motion

Newton’s Laws for Rotation

at

In e t

1st part

[s–2]

3rd part

[N m]

2nd part

[kg m2]

Krot = (1/2) I w2 (K = (1/2) m v2)

5

Rotational Motion

Angular Quantities

Kinematical variables to describe the rotational motion:

Angular position, velocity and acceleration

6

Rotational Motion

Angular Quantities

Kinematical variables to describe the rotational motion:


)rad/s(lim

)rad/s(lim

)rad(

2

0

0

dt

d

ave

t

dt

d

ave

t

R

l

t

t

w

a

wa

q

w

qw

q

q = q2 – q1

l = arc of length

7

Rotational Motion

Vector Nature of

Angular Quantities Kinematical variables to describe the rotational motion:


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

Vector natures!

)rad/s(

)rad/s(

)rad(

2 k̂

dt

dk̂

k̂ dt

dk̂

R

l

wa

qw

q

x

y

z

R.-H. Rule

>0 or <0

8

Rotational Motion

Vector Nature of

Angular Quantities Kinematical variables to describe the rotational motion:


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

Vector natures!

9

Rotational Motion

Vector Nature of

Angular Quantities

10

Rotational Motion

Kinematic Equations

awq avx , , : C o n v e r s i o n

11

Rotational Motion

Example 1 (a = constant) A bicycle wheel has an initial angular velocity of 1.50 rad/s. Its

angular acceleration is constant and equal to 0.300 rad/s2.

a) What is its angular velocity at t = 2.50 s?

b) Through what angle has the wheel turned between t = 0 and

t = 2.50 s? y

x

12

Rotational Motion





t = 2.50 s?

s.rad 252

s) 502)(srad 3000(srad 501

a)

2

0

.

...

tαωω zzz

rad. 694

s) 502)(srad 3000(s) 502)(srad 501(

21 b)

22

21

2

0

.

....

tαtωθ zz

y

x

13

Rotational Motion





t = 2.50 s?

s.rad 252

s) 502)(srad 3000(srad 501

a)

2

0

.

...

tαωω zzz

rad. 694

s) 502)(srad 3000(s) 502)(srad 501(

21 b)

22

21

2

0

.

....

tαtωθ zz

y

x

14

Rotational Motion

What is the resultant angular velocity of

the wheel, as seen by an outside observer?

r a d / s 0.5 0w h e e l w

r a d / s 0.3 5t u r n t a b l e w

turntable

wheel

x

y

Let’s see how move. 15

Rotational Motion

y

z

x : out of page

r a d / s 0.3 5t u r n t a b l e w

r a d / s 0.5 0w h e e l w

t u r n t a b l ew h e e lr e s u l t a n t www

Easy Way: Vector Analysis

18

Rotational Motion

Linear and Angular Quantities

RR

R

Ra

Ra

d

dR

d

da

R

Rl

dωt R

l θ

222

rad

tan

tan

)( (3)

(2)

)(

)(like (1)

ww

a

w

w

w

v

v

v

v

tt

t

t

atan

arad

(Increase/decrease the speed..)

(Maintain the circular motion..)

l = arc of length

19

Rotational Motion

Physics in Throwing a Discus

R = 80 cm

w = 10.0 rad/s

a = 50.0 rad/s2

atan = ?

arad = ? 23

Rotational Motion

“R” from the Axis (O)

Solid Disk Solid Cylinder

rR rR 24

Rotational Motion

Example 1x (a = constant) (1) What is the angular speed of rotation of the Earth?

1 rotation (2p rad) per 24 hours (1440 min)

rotational speed in rotation per minute (rpm) or rad/s

(2) An old 33 1/3 rpm record player starts from rest and

reaches operating speed in 2.00 seconds. Through what

angle did it turn in those 2.00 seconds?

t = 2.00 s, w0 = 0, w = 33 1/3 rpm

a from Eq. (2)

Use Eq. (1) with q0 = 0

(3) A computer hard drive rotates at 5400 rpm. What

angular acceleration will get it up to speed in just 150

revolutions starting from rest?

Use Eq. (3) with q – q0 = 150 rev. = ? rad )(2 ( 3 ) 0

2

0

2 qqaww

t

t t

aww

awqq

0

2

00

(2)

2

1 (1)

26

Rotational Motion

Example 1x (cont’d)

(4) A hard drive reaches 5400 rpm in 3.20 seconds. What

was the average angular speed assuming constant

angular acceleration?

(5) A dentist’s drill accelerates to 1800 rpm in 2.50 seconds.

what is its angular acceleration?

(6) The angular velocity changes from 47.0 rad/s to –47.0

rad/s in 2.00 seconds. What is the angular acceleration?

28

Rotational Motion

Example 2

A grinding wheel turns at a constant angular acceleration

of 60.0 rad/s2 from 24.0 rad/s for 2.00 sec. Then, a

circuit breaker trips. It turns through 432 rad as it coasts

to a stop at a constant angular acceleration.

Find:

(a) the total angle between t = 0

and the time it stopped;

(b) the time it stopped;

(c) the angular acceleration as it

slowed down.

Also sketch q-t graph.

30

Rotational Motion

Visualization via Graph

432 rad

q1 rad

2 sec T sec

t

q

a = ? rad/s2

a = 60.0 rad/s2

w0 = 24.0 rad/s

?

? 31

Rotational Motion

Example 2 - Solutions

35

Rotational Motion

Problem 3: (25 points)

A grinding wheel turns at a varying angular acceleration of a(t) = [30.0 rad/s3] t for 2.00

sec. Assume the initial angular speed of 20.0 rad/s. Then, a circuit breaker trips. It turns

through 400 rad as it coasts to a stop at a constant angular acceleration.

a. (5 pts) Find the total angle (qtotal) between t = 0 and the time it stopped.

b. (10 pts) Find the time (ttotal) it stopped. Find the angular acceleration as it slowed down.

c. (10 pts) Sketch qt, w-t, a-t graphs.

Practice Problem 1

36

Rotational Motion

Practice Problem 2 A solid cylinder (radius R = 2 m and height H = 5 m) turns at a

constant angular acceleration of 60.0 rad/s2 from 24.0 rad/s for

2.00 sec. Then, a circuit breaker trips. It turns through 432 rad

as it coasts to a stop at a constant angular acceleration.

(a) Find the total angle between t = 0

and the time it stopped.

(b) Find the time it stopped.

(c) Find the angular acceleration

as it slowed down.

(d) Find the speed (v) of point P

at t = 2.00 sec.

(e) Sketch the motion of point P

in v-t graph. Also sketch q-t graph.

H 0.4H

w

37

Rotational Motion

Newton’s Laws for Rotation

at

In e t

1st part

[s–2]

3rd part

[N m]

2nd part

[kg m2]

Krot = (1/2) I w2 (K = (1/2) m v2)

38

Rotational Motion

Rotational Kinetic Energy and Inertia

particles) of group afor particle) singlefor

inertia of(moment inertia of(moment

(b) (a)

2

1)(

2

1

2

1

2

1

2

1

)(2

1 )(

2

1

2

1

222

321

2

3

2

2

2

1

2222

2

ii

2

i

ii

Rm I R mI

RmvmmmvmvmvmK

mRRmmvK

w

ww

m1

m2

m3 39

Rotational Motion

Rotational K.E. about B-C?

w = 4.0 rad/s Krot = (1/2) I w2

40

Rotational Motion

Which one has a bigger I?

Fp

Fp

41

Rotational Motion

Rotational Inertia

42

Mechanical Energy Conservation

K = Km + KM

Krot = (1/2) I w2 (K = (1/2) m v2)

Rotational Motion

Solid disk (M, R0)

m

44

Mechanical Energy Conservation

Rotational Motion

Two blocks with masses m1 = 35.0 kg and m2 = 38.0 kg are

connected by a rope that hangs over a pulley. The pulley is a

uniform disk of radius R0 = 0.300 m and mass M = 4.80 kg.

Initially m1 is on the ground and m2 rests h = 2.50 m above the

ground. Then the system is released. Find the speed of m2 just

before it strikes the ground. Use conservation of mechanical energy.

Assume that the pulley bearing is frictionless and

that the rope does not slip on the pulley rim.

45

M

Example 1

Rotational Motion

Parallel-axis Theorem

Icm = ½ MR02

d

2

c m M dII

Idisk = Icylinder

47

Rotational Motion


d

2

0

2

0

2

0

2

2

3

2

1

MR

MRMR

MdII

cm

48

Rotational Motion


d1

d2

21 III

49

Rotational Motion


d1

2

0

2

0

2

0

2

0

2

0

21

11

32

1

2

1

22

21

MR

RMMR

RMMR

III

d

d

d2

50

Rotational Motion

22

2

2

fg3

1

41 2

1

2l M

lMl M

lM I I


51

Rotational Motion

2(a) Express the moment of inertia of the array of point

objects about the y-axis in terms of m, M, X1, X2, and/or

Y.

X1 X2

Y

52

Rotational Motion

2(b) Consider a helicopter rotor blade as a long thin rod. If

each of the three blades is 3.75 m long and has a mass of

160 kg, calculate the moment of inertia of the three blades

about the axis of the rotation.

53

Rotational Motion

2(c) A meter stick (mass M = 0.500 kg and length L = 1.00 m) is

hung from the wall by means of a metal pin through the hole,

and used as a pendulum. Express the moment of inertia of the

stick about the pin (= the axis of the rotation) in terms of M, L,

and x.

54

Rotational Motion

2(d) A door (solid rectangular thin plate) of mass M = 15.0 kg

is free to rotate on about hinge line. Calculate the moment of

inertia of the door about the hinge line.

H

W

55

Rotational Motion

2(e) A solid disk (mass M = 3.00 kg and radius R = 20.0 cm) is

hung from the wall by means of a metal pin through the hole,

and used as a pendulum. Calculate the moment of inertia of the

disk about the pin (= the axis of the rotation).

56

Rotational Motion

2(f) A ball (solid sphere) of mass M and radius R on the end of

a thin rod (mass m and length l). Express the moment of

inertia of the system of the rod and the ball about the A-B axis

(thin rod; mass m and length l) in terms of M, R, m, and l.

l

m

M R

m

l

Hin

t

57

58

Moment-of-Inertia Calculation

[Key Concept] We divide the rod into many small segments (particles). I Itotal = S[I] Integral

Figure 9.24

Uniform Solid Cone


59

[Key Concept] We divide the cone into many thin, solid disks. I Itotal = S[I] Integral

60

Uniform Solid Cone


62


[Key Concept] We divide the solid shape into many known elements. I Itotal = S[I] Integral

Documents

Chap. 9: Rotational Motionpeople.physics.tamu.edu/kamon/teaching/phys218/slide/... · 2013-04-01 · Rotational Motion Problem 3: (25 points) A grinding wheel turns at a varying angular