PHY 113 A Fall 2012 -- Lecture 17 110/10/2012

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

Plan for Lecture 17:

Chapter 10 – rotational motion

1. Angular variables

2. Rotational energy

3. Moment of inertia

PHY 113 A Fall 2012 -- Lecture 17 210/10/2012

PHY 113 A Fall 2012 -- Lecture 17 310/10/2012

Angular motion

angular “displacement” q(t)

angular “velocity” angular “acceleration”

dtd(t) ωα

dtd(t) θω

“natural” unit == 1 radian

Relation to linear variables: sq = r (qf-qi)

vq = r w

aq = r a

s

PHY 113 A Fall 2012 -- Lecture 17 410/10/2012

Special case of constant angular acceleration: a = a0:

w(t) = wi + a0 t

q(t) = qi + wi t + ½ a0 t2

( w(t))2 wi2 + 2 a0 (q(t) - qi )

w

r1v1=r1w

v2=r2w

r2

PHY 113 A Fall 2012 -- Lecture 17 510/10/2012

A wheel is initially rotating at a rate of f=30 rev/sec.

smRvR

πfω

/247.94)0.5m)(rad/s 495.188( 0.5m? radius aat wheel theof

rim on thedot a of speed theisWhat rad/s 188.495 rad/s )30(22

elocity?angular v theisWhat

w

R

PHY 113 A Fall 2012 -- Lecture 17 610/10/2012

A wheel is initially rotating at a rate of f=30 rev/sec. Because of a constant angular deceleration, the wheel comes to rest in 3 seconds.

22

2

/42.31)0.5m)(rad/s 83.62(

0.5m? radius aat wheel theofrim on thedot a ofon decelerati theisWhat

rad/s 83.26 3

rad/s )30(2320

on?deceleratiangular theisWhat

smRa

R

ssπf

a

a

R

PHY 113 A Fall 2012 -- Lecture 17 710/10/2012

Example: Compact disc motion

In a compact disk, each spot on the disk passes the laser-lens system at a constant linear speed of vq = 1.3 m/s.

w1=vq/r1=56.5 rad/s

w2=vq/r2=22.4 rad/s

What is the average angular acceleration of the CD over the time interval Dt=4473 s as the spot moves from the inner to outer radii?

a = (w2-w1)/Dt =-0.0076 rad/s2

w1

w2

PHY 113 A Fall 2012 -- Lecture 17 810/10/2012

Object rotating with constant angular velocity (a = 0)

w

v=0

v=Rw

Kinetic energy associated with rotation:

“moment of inertia”

iii

iii

iii

rmI

IrmvmK

2

22

1222

122

1

:where

;ωω

R

PHY 113 A Fall 2012 -- Lecture 17 910/10/2012

Moment of inertia: i

iirmI 2

iclicker exercise: Which case has the larger I? A. a B. b

PHY 113 A Fall 2012 -- Lecture 17 1010/10/2012

Moment of inertia: i

iirmI 2

22MaI 22 22 mbMaI

PHY 113 A Fall 2012 -- Lecture 17 1110/10/2012

Note that the moment of inertia depends on both

a) The position of the rotational axisb) The direction of rotation

m m

d d

I=2md2

m m

d d

I=m(2d)2=4md2

PHY 113 A Fall 2012 -- Lecture 17 1210/10/2012

iclicker question:

Suppose each of the following objects each has the same total mass M and outer radius R and each is rotating counter-clockwise at an constant angular velocity of w=3 rad/s. Which object has the greater kinetic energy?

(a) (Solid disk) (b) (circular ring)

PHY 113 A Fall 2012 -- Lecture 17 1310/10/2012

Various moments of inertia:

solid cylinder:

I=1/2 MR2

solid sphere:

I=2/5 MR2

solid rod:

I=1/3 MR2

R

RR

PHY 113 A Fall 2012 -- Lecture 17 1410/10/2012

Calculation of moment of inertia:

Example -- moment of inertia of solid rod through an axis perpendicular rod and passing through center:

2222

31

22MRdrr

RMrdr

RMrmI

R

R

R

Riii

R

PHY 113 A Fall 2012 -- Lecture 17 1510/10/2012

Note that any solid object has 3 moments of inertia; some times two or more can be equal

ji

k

iclicker exercise:Which moment of inertia is the smallest? (A) i (B) j (C) k

PHY 113 A Fall 2012 -- Lecture 17 1610/10/2012

iclicker exercise:Three round balls, each having a mass M and radius R, start from rest at the top of the incline. After they are released, they roll without slipping down the incline. Which ball will reach the bottom first?

AB C

2MRI A

22 5.021 MRMRIB

22 4.052 MRMRIC

PHY 113 A Fall 2012 -- Lecture 17 1710/10/2012

22

21

21

:object rolling ofenergy kinetic Total

CM

CMrollingtotal

MvI

KKK

w

CMvRdtdR

dtds

dtd

wq

qw

: thatNote

( )

22

222

21

21

21

CM

CM

CMrollingtotal

vMRI

MvRRI

KKK

w

PHY 113 A Fall 2012 -- Lecture 17 1810/10/2012

iclicker exercise:Three round balls, each having a mass M and radius R, start from rest at the top of the incline. After they are released, they roll without slipping down the incline. Which ball will reach the bottom first?

AB C

2MRI A

22 5.021 MRMRIB

22 4.052 MRMRIC