Rolling

cmv

Rolling

cmv

s

s

Rs

Rdtd

dtds

Rvcm

Rolling Condition – must hold for an object to roll without slipping.

R

Rolling

One way to view rolling is as a combination of pure rotation and pure translation.

Pure Rotation

cmv

cmv

cmv

cmv

cmv

2

Pure Translation Rolling

cmv

cmv

Rvcm

The point that is in contact with the ground is

not in motion with respect to the ground!

Rolling

cmv

2

Rolling

cmv

The point that is in contact with the ground is

not in motion with respect to the ground!

Since the bottom point is at rest with respect to the ground, static friction applies if any

friction exists at all. Static friction does not dissipate energy.

However, there usually is rolling friction caused by the deformation of the object and surface as well as the loss of pieces of the

object. Rolling friction does dissipate energy.

Rolling

If the disk is moving at constant speed, there is no tendency to slip at the contact point and so there is no frictional force.

If, however, a force acts on the disk, like when you push on a bike pedal, then there is a tendency to slide at the point of contact so a frictional force acts at

that point to oppose that tendency.

Rolling

Just as rolling motion can be viewed as a combination of pure rotation and pure translation, the kinetic energy of a rolling object can be viewed as a combination of pure

rotational kinetic energy and pure translational kinetic energy.

ntranslatio of rotation of rolling of kkk EEE

22

21

21

mvIE cmk rolling of

Pure Rotation

Pure Translation

Rolling

.100

0214

sm.

kg. cm

of speed constant

a withtable horizontal a across rolls mass and radius of hoop A 1.

a. What is the kinetic energy of the hoop?

22

21

21

MvIEk

ButRvcm

Rvcm

Note: the v in the Ek equation is vcm

2MRIhoop

22

2

21

21

MvRv

MREk

Rolling

.100

0214

sm.

kg. cm

of speed constant

a withtable horizontal a across rolls mass and radius of hoop A 1.

a. What is the kinetic energy of the hoop?

22

21

21

MvMvEk

2MvEk

210.00.2 smkgEk

JEk2100.2

Rolling

.100

0214

sm.

kg. cm

of speed constant

a withtable horizontal a across rolls mass and radius of hoop A 1.

b. What percentage of the kinetic energy is associated with rotation and what percentage with translation?

22

21

21

MvMvEk

%50% rotation %50% ntranslatio

Rolling

.100

0214

sm.

kg. cm

of speed constant

a withtable horizontal a across rolls mass and radius of hoop A 1.

n?translatio withpercentage whatand rotation withassociate isenergy kinetic

its of percentage what,52

sphere solid a instead is object the If c.

2MRI

22

21

21

MvIEk

ButRvcm

Rvcm

2

52

MRIsphere

22

2

21

52

21

MvRv

MREk

Rolling

.100

0214

sm.

kg. cm

of speed constant

a withtable horizontal a across rolls mass and radius of hoop A 1.

22

21

51

MvMvEk

%29% rotation

%71% ntranslatio

n?translatio withpercentage whatand rotation withassociate isenergy kinetic

its of percentage what,52

sphere solid a instead is object the If c.

2MRI

22

105

102

MvMvEk

2

107

MvEk

100

107

102

% rotation

100

107

105

% ntranslatio

Rolling

2. A hoop, a disk, and a solid sphere with identical masses and radii roll down an incline of length 2.4 m and angle 35º.

a. Using dynamics (Newton’s Laws), what is the acceleration of the center of mass of each object as it rolls down the incline?

35

gFgyF

gxF

35

NF

fsF gyN FF

fsgx FF

Rolling

2. A hoop, a disk, and a solid sphere with identical masses and radii roll down an incline of length 2.4 m and angle 35º.

a. Using dynamics (Newton’s Laws), what is the acceleration of the center of mass of each object as it rolls down the incline?

xx amF

maFF fsgx

yy amF

0 gyN FF

cosmgFN

I

IRFfs

FN and Fg exert no torque since they act through the axis of

rotation (cm)

ButRa

Ra

2R

IaFfs

maFmg fs sin

Rolling

2. A hoop, a disk, and a solid sphere with identical masses and radii roll down an incline of length 2.4 m and angle 35º.

a. Using dynamics (Newton’s Laws), what is the acceleration of the center of mass of each object as it rolls down the incline?

maR

Iamg 2sin

2sinR

Iamamg

aR

Immg

2sin

2

sin

R

Im

mga

21

sin

mR

Ig

a

Rolling

2. A hoop, a disk, and a solid sphere with identical masses and radii roll down an incline of length 2.4 m and angle 35º.

a. Using dynamics (Newton’s Laws), what is the acceleration of the center of mass of each object as it rolls down the incline?

21

sin

mR

Ig

a

But2mRIhoop 2

21

mRIdisk 2

52

mRIsphere

2

2

1

sin

mR

mR

gah

2

2

21

1

sin

mR

mR

gad

2

2

52

1

sin

mR

mR

gas

Rolling

2. A hoop, a disk, and a solid sphere with identical masses and radii roll down an incline of length 2.4 m and angle 35º.

a. Using dynamics (Newton’s Laws), what is the acceleration of the center of mass of each object as it rolls down the incline?

11sin

gah

21

1

sin

g

ad

52

1

sin

g

as

sin21

gah sin32

gad sin75

gas

28.2s

mah 27.3s

mad 20.4s

mas

Rolling

2. A hoop, a disk, and a solid sphere with identical masses and radii roll down an incline of length 2.4 m and angle 35º.

b. In what order would the hoop, disk, and sphere reach the bottom of the incline?

sin21

gah

sin32

gad

sin75

gas

?t

0ov

mLxx o 4.2

2

21

tatvxx oo

aL

t 2

sin21

2

g

Lth

sin32

2

g

Ltd

sin75

2

g

Lts

sin4

gL

th sin

3g

Ltd

sin514g

Lts

sth 31.1 std 13.1 sts 09.11#2#3#

Rolling

2. A hoop, a disk, and a solid sphere with identical masses and radii roll down an incline of length 2.4 m and angle 35º.

a. Using energy principles , what is the velocity of the center of mass of each object as it reaches the bottom of the incline?

35 L

iy

sinLyi

Rolling

2. A hoop, a disk, and a solid sphere with identical masses and radii roll down an incline of length 2.4 m and angle 35º.

a. Using energy principles, what is the velocity of the center of mass of each object as it reaches the bottom of the incline?

int0 E

gk EE 0

gigfkikf EEEE 0

gikf EE

iff mgymvI 22

21

21

But

sinLyi Rvcm

Rv

Rolling

2. A hoop, a disk, and a solid sphere with identical masses and radii roll down an incline of length 2.4 m and angle 35º.

a. Using energy principles, what is the velocity of the center of mass of each object as it reaches the bottom of the incline?

sin21

21 2

2

LmgmvR

vI f

f

sin2222 mgLmvv

R

Iff

sin2222 gLvv

mR

Iff

sin21 22 gLv

mR

If

Rolling

2. A hoop, a disk, and a solid sphere with identical masses and radii roll down an incline of length 2.4 m and angle 35º.

a. Using energy principles, what is the velocity of the center of mass of each object as it reaches the bottom of the incline?

2

2

1

sin2

mR

IgL

v f

21

sin2

mR

IgL

v f

Rolling

2. A hoop, a disk, and a solid sphere with identical masses and radii roll down an incline of length 2.4 m and angle 35º.

a. Using energy principles, what is the velocity of the center of mass of each object as it reaches the bottom of the incline?

21

sin2

mR

IgL

v f

But2mRIhoop 2

21

mRIdisk 2

52

mRIsphere

2

2

1

sin2

mR

mR

gLv fh

2

2

21

1

sin2

mR

mR

gLv fd

2

2

52

1

sin2

mR

mR

gLv fs

Rolling

2. A hoop, a disk, and a solid sphere with identical masses and radii roll down an incline of length 2.4 m and angle 35º.

a. Using energy principles, what is the velocity of the center of mass of each object as it reaches the bottom of the incline?

11sin2

gLv fh

21

1

sin2

gL

v fd

52

1

sin2

gL

v fs

singlv fh sin34

glv fd sin7

10glv fs

smv fh 7.3 s

mv fd 2.4 smv fs 4.4

Rolling

2. A hoop, a disk, and a solid sphere with identical masses and radii roll down an incline of length 2.4 m and angle 35º.

b. In what order would the hoop, disk, and sphere reach the bottom of the incline?

?t

0ov

mLxx o 4.2

tvvxx ofo

21

sth 31.1 std 13.1 sts 09.11#2#3#

singlv fh

sin34

glv fd

sin7

10glv fs

f

o

vxx

t 2

sin2

gLL

th sin

34

2

gL

Ltd

sin7

102

gL

Lts