LECTURE 2- ROLLING MOTIONPhys 124H- Honors Analytical Physics IBChapter 11Professor Noronha-Hostler

IF YOU’RE NEW THIS WEEK

Read last week’s lecture on http://www.physics.rutgers.edu/~jn511/PHY124H.html

Make sure you have homework (Wiley+) set up (frbaratta@wiley.com)

AFTER reading the website/Lecture 1, come see me in office hours if you have further questions

Make sure I have anything needed from ODS.

Gradebook is usual Rutgers credentials (NETID and password)

TODAY’S OBJECTIVES

Rolling wheel

Torque as a vector

Friction

Angular momentum

WHICH REACHES THE BOTTOM FIRST?

A.) Hollow sphere of Mass M

B.) Solid Sphere of Mass M

C.) Not enough info

D.) Solid disk of Mass M

E.) Hoop of Mass M

C.) Not enough info

WHICH REACHES THE BOTTOM FIRST?

Both a hoop and solid cylinder have the same

I =12

MR2

K =12

Iω2 = const If they are all the same mass only!

What would happen if we doubled the mass of one of them?

ROLLING STONES (WHEELS)

ROLLING WHEEL (NO FRICTION)

s = θR

Linear speed at the center of mass

vcom = ωR

Translation motion

TRANSLATION+ROTATIONAL

At point P the velocity is 0!

What is in figure (b)?ω

KINETIC ENERGY OF ROTATION

K =12

Icomω2 +12

Mv2comKinetic energy of rolling wheel

Where does this come from?

K =12

Ipw2Kinetic Energy at point p

New perspective, wheel is rotating around point P

KINETIC ENERGY OF ROTATION

K =12

Icomω2 +12

Mv2comKinetic energy of rolling wheel

K =12

Ipω2 But we want the motion at the com

Parallel-axis theorem Ip = Icom + MR2

K =12

Icomω2 +12

MR2ω2

KINETIC ENERGY OF ROTATION

K =12

Icomω2 +12

Mv2comKinetic energy of rolling wheel

What is missing? vcom = ωr

K =12

Icomω2 +12

MR2ω2

K =12

Icomω2 +12

Mv2com

2 TYPES OF KINETIC ENERGY

K =12

Icomω2

rotational

+12

Mv2com

translational

Iclicker #1. A circular hoop rolls without slipping on a flat horizontal surface. Which one of the following is necessarily true?

a) All points on the rim of the hoop have the same speed.

b) All points on the rim of the hoop have the same linear velocity.

c) Every point on the rim of the wheel has a different linear velocity.

d) All points on the rim of the hoop have acceleration vectors that are tangent to the hoop.

e) All points on the rim of the hoop have acceleration vectors that point toward the center of the hoop.

c) Every point on the rim of the wheel has a different linear velocity.

A bicycle wheel of radius 0.70 m is turning at an angular speed of 6.3 rad/s as it rolls on a horizontal surface without slipping. What is the linear speed of the wheel?

a) 1.4 m/s

b) 28 m/s

c) 0.11 m/s

d) 4.4 m/s

e) 9.1 m/s

d) 4.4 m/s

BICYCLE LINEAR VELOCITY

v = ωr

v = 6.3 rad/s ⋅ 0.70 m

v = 4.4 m/s

Radians can be thought of as an “invisible” unit i.e. if we used

.

rads

⋅ ft =fts

rads

⋅ m =ms

A solid cylinder is rotating about an axis that passes through the center of both ends. The radius of the cylinder is r. At what angular speed must this cylinder rotate to have the same total kinetic energy that it would have if it were moving horizontally with a speed v without rotation?

A.)

B.)

C.)

D.)

E.)

ω

ω =v2

2rω =

vr

2

ω =vr

ω =v2r

ω =v2

r2

ω =vr

2

r

Versus

r

Rotating around an axis Moving without rotating

v

ωI =

12

Mr2

Krot =12

Iω2 Kslide =12

Mv2

12

Iω2 =12

Mv2

Krot = Kslide

I =12

Mr2Iω2 = Mv2

12

Mr2ω2 = Mv2

ω2 = 2v2

r2

ω =vr

2

WHAT IS NEEDED FOR WHEELS TO ROTATE?

ROLLING+FRICTION

No sliding? Only static friction fs

Wheels need friction to accelerate , which leads to a com

But, wheels also tend to slip, then there is kinetic friction

α

P

a com

fs

Smooth rollingfk

This is called smooth rolling.

acom = αR

ROLLING DOWN A HILL+FRICTION

https://www.youtube.com/watch?v=XjMDvptIbP0

Start at 46 seconds. What happens at ~56 seconds?

ROLLING DOWN A RAMP

Newton’s 2nd Law F = Ma

fs − Mg sin θ = Macom,x

θ

R

F N

F gF g cos θ

F g sin θθ

f sx+

y+

ROLLING DOWN A RAMP

Newton’s 2nd Law (angular) τnet = Icomα

Torque τ = r⊥F

Rfs = Icomα

θ

R

F N

F gF g cos θ

F g sin θθ

f sx+

y+

DIRECTION OF ACCELERATION

Is the wheel rolling clockwise or counterclockwise?

With smooth rolling, wheel goes counter clockwise, so α > 0But, we have acom < 0 so acom = − α/R

θ

R

F N

F gF g cos θ

F g sin θθ

f sx+

y+

Demo

ROLLING DOWN RAMP+FRICTION

Rfs = Icomα

Collecting the equations

fs = Icomα/R

fs = − Icomacom,x /R2

acom,x = −g sin θ

1 + Icom/MR2

Recall

fs − Mg sin θ = Macom,x

TORQUE AS A VECTOR (DEMO)

τ = r × F

c = a × bRecall where |c | = |b | |c |sin ϕ

Clockwise rotation points into the page

Counter clockwise points out of the page

⊗

⊙

SUMMARY OF EQUATIONS

What is the analogy of momentum?

ANGULAR MOMENTUM L

l = r × p Recall p = m v

l = m ( r × v )Before, we thought of p when doing collisions.

Where does a baseball batter what to hit a ball?

or

ANGULAR MOMENTUM AS A VECTOR

l = m ( r × v ) Magnitude

Direction

| l | = m | r | | v |sin ϕ

Right-hand ruleSmallest angle between

r and p.

UNDERSTANDING FIGURE SKATING SPINS

RELATING TORQUE AND ANGULAR MOMENTUM

Newton’s second law F net =d pdt

How do you think angular momentum will look like?

τnet =d ldt

WHAT ABOUT MANY PARTICLES?

12

3 L = l1 + l2 + … + ln =n

∑i

l i

n

∑i

τnet,i =n

∑i

d l i

dt=

d Ldt

This would include internal and external torques. But what does Newton’s third imply for internal torques?

For every action there is an equal and opposite reaction, so we only need to consider external torques.

τnet =d Ldt

CONSERVATION OF ANGULAR MOMENTUM For an isolated system (no external forces)

L = const

L i = L f

Demo (chair)

EXAMPLE

1

2

3

Find the moment of Inertia

I = ∑i

mir2i

dd

Give the stick that is rotating around point 1 where all three balls are of mass, m, and the stick’s

mass is negligible.

I = m ⋅ 0 + m ⋅ d2 + m ⋅ (2d)2 = 5md2

EXAMPLE

1

2

3

Find the angular momentum of particle 2.d

dGive the stick that is rotating around point 1

where all three balls are of mass, m, and the stick’s mass is negligible.

L2 = I2ωω

I2 = md2

L2 = md2ω

EXAMPLE

1

2

3

What is the total angular momentum?d

dGive the stick that is rotating around point 1

where all three balls are of mass, m, and the stick’s mass is negligible.

L2 = I2ωω

L = 0 + md2ω + 4md2ω = 5md2ω

EXAMPLE

1

2

3

If the ball 3 falls off, how fast with the stick rotate?d

dGive the stick that is rotating around point 1

where all three balls are of mass, m, and the stick’s mass is negligible.

ω Li = Lf

Li = 5md2ωi Lf = Ifωf

If = md2ωf = 5md2ωi/If

ωf = 5ωi

GYROSCOPE (DEMO)

Precession: When the axis of rotation is no longer fixed

GYROSCOPE (DROP A SPINNING TOP)

Why doesn’t it fall over?

τ =d Ldt

τ = Mgr sin 90∘ = Mgr

Rapidly spinning, begins to rotate horizontally about its vertical axis

L i ≠ 0 Drop doesn’t change angular momentum

GYROSCOPE CONTINUED

L = IωWe know that

andτ =

d Ldt

d L = τdt

For the rapidly spinning gyroscope, is fixed, so only direction changes.

| L |

d L = τdt τ = Mgr

d L = Mg rdt

GYROSCOPE CONTINUED

d L = Mg rdtTo understand the change in angle, we use Ldϕ = dL

dϕ =dLL

=Mg rdt

L=

Mg rdtIω

Ω ≡dϕdt

=Mg rIω

Precession rate

Ω ⇑ when ω ⇓

A hollow cylinder of mass M and radius R rolls down an inclined plane. A block of mass M slides down an identical inclined plane. Complete the following statement: If both objects are released at the same time,

a) the cylinder will reach the bottom first.

b) the block will reach the bottom first.

c) the block will reach the bottom with the greater kinetic energy.

d) the cylinder will reach the bottom with the greater kinetic energy.

e) both the block and the cylinder will reach the bottom at the same time.

b) the block will reach the bottom first.

WHAT IS THE DIRECTION OF THE EARTH’S ORBITAL ANGULAR MOMENTUM AS IT SPINS ABOUT ITS AXIS?

a) north

b) south

c) east

d) west

e) radially inward

a) north

NEXT WEEK

Equilibrium and elasticity

Viscosity

Center of gravity