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PHY131H1F - Class 17

Today:

Rolling Without Slipping

Rotational Energy

• Reminder: Term test tomorrow from 6:10 to 7:30 pm.

• It will be based on the Uncertainties Reading plus Wolfson

Chapters 5-9 and Sections 10.1-10.3.

• The room you will write in is based on your last name:

• A – I: EX100

• J – R: EX200

• S – Tr: EX300

• Ts – Xi: EX310

• Xu – Z: EX320

Review of Section 10.3

Which pencil has the largest

rotational inertia?

A. The pencil rotated around an

axis passing through it.

B. The pencil rotated around a

vertical axis passing through

centre.

C. The pencil rotated around

vertical axis passing through

the end.

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Class 17 Preclass Quiz on MasteringPhysics

This was due this morning at 8:00am

84% of students got: A rotating object has some rotational

kinetic energy. If its angular speed is doubled, but nothing

else changes, the rotational kinetic energy increases by a

factor of 4.2

rot2

1IK

Class 17 Preclass Quiz on MasteringPhysics

This was due this morning at 8:00am

78% of students got: A marble of radius 1 cm rolls with a

speed of 3 cm/s. Its angular speed is 3 rad/s.

𝜔 =𝑣

𝑟

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Class 17 Preclass Quiz on MasteringPhysics

This was due this morning at 8:00am

79% of students got: Starting from rest, a solid sphere rolls

without slipping down an incline plane. At the bottom of the

incline, what does the angular velocity of the sphere depend

upon?

Check all that apply.

The angular velocity depends upon the length of the incline.

The angular velocity depends upon the mass of the sphere.

The angular velocity depends upon the height of the

incline.

The angular velocity depends upon the radius of the

sphere.

Class 17 Preclass Quiz Student Comments

“so apparently, I will have three midterms in one week after fall break....容我做一个痛苦的表情”

“I went to youtube to watch Masterchef. In the

search box, instead of typing MasterChef I typed

Mastering Physics. This happened TWICE. WHAT

HAV U DONE TO ME??”

“Do you think I could calculate the angular velocity of the tears rolling down my face right now?”

Harlow note: I understand it is a very busy time for

all of you. Please take care of your health first and

foremost: remember, in 5 weeks you will be on

winter break and this semester will be behind you!

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Class 17 Preclass Quiz Student Comments

“I don't understand how we get i, the rotational

inertia. And how does a hoop have more rotational

inertia than a disk?”

Let’s ask you this: There are two identical solid

disks. A big hole is cut out of one of them, so it

becomes a hoop. Which has the larger rotational

inertia?

A: The solid disk B: The hoop C: neither

Class 17 Preclass Quiz Student Comments

“I don't understand how we get i, the rotational inertia. And

how does a hoop have more rotational inertia than a disk?”

Let’s ask you this: There are two identical solid disks. A big

hole is cut out of one of them, so it becomes a hoop. Which

has the larger rotational inertia?

A: The solid disk B: The hoop C: neither

• Remember, the formula for rotational inertia is a sum.

• So, if an object can be built by combining object 1 with rotational

inertia I1 plus object 2 with rotational inertia I2, then the total

rotational inertia is I1 + I2.

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Class 17 Preclass Quiz Student Comments

“I don't understand how we get i, the rotational inertia. And

how does a hoop have more rotational inertia than a disk?”

If both the hoop and disk have the same mass, and same

outer radius, then this hoop must be much more dense than

the disk. For example, the hoop might be made of metal and

the disk of wood. In this case, since more mass is

concentrated away from the rotation axis for the metal hoop,

then it has a higher rotational inertia.

𝐼 =1

2𝑀𝑅2 𝐼 = 𝑀𝑅2

From a student email this morning (at 1:30am)

“Other than the integration formula for x to the

power of an integer, are there any other integration

formulas that we need to memorize?”

Harlow answer: No.

Also, there will be no 3-D integrals on the test – ie

no spheres, cones, etc.

You might note already that I did not include an

integral table on the test, which means the only

integral you might get is this one:

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10.5: Rolling without slipping

No matter what the

speed, four points on this

car are always at rest!

Which points? The

bottoms of the four tires!

A wheel rolls much like

the treads of a tank.

The bottom of the wheel

is at rest relative to the

ground as it rolls.

Rolling without slipping

S’ frame: the axle

S frame: the ground

ω

The wheel rotates with angular speed ω.

The tangential speed of a point on the rim is v = ωr,

relative to the axle.

In “rolling without slipping”, the axle moves at

speed v. This is the S’ frame.

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Rolling without slipping

S’ frame: the axle S frame: the ground

Rolling without slipping

ω

The wheel rotates with angular speed ω.

If your car is accelerating or decelerating

or turning, it is static friction of the road on

the wheels that provides the net force which

accelerates the car

The axle moves with linear speed v = ωr.,

where r is the radius of the wheel.

v = ωr

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Rolling Without Slipping

Under normal driving

conditions, the portion of the

rolling wheel that contacts

the surface is stationary, not

sliding

In this case the speed of

the centre of the wheel is:

where C = circumference [m] and T = Period [s]

𝑣 =𝐶

𝑇

Discussion Question• The circumference of the tires on your car is

0.9 m.

• The onboard computer in your car measures

that your tires rotate 10 times per second.

• What is the speed as displayed on your

speedometer?

A. 0.09 m/s

B. 0.11 m/s

C. 0.9 m/s

D. 1.1 m/s

E. 9 m/s

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The “Rolling Without Slipping”

ConstraintsWhen a round object rolls without slipping, the distance the axis, or centre of mass, travels is equal to the change in angular position times the radius of the object.

s = θR

The speed of the centre of mass is

v = ωR

The acceleration of the centre of mass is

a = αR

Rotational Kinetic Energy

A rotating rigid body has kinetic energy because all atoms in the object are in motion. The kinetic energy due to rotation is called rotational kinetic energy.

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Example: A 0.50 kg basketball rolls along the ground at

1.0 m/s. What is its total kinetic energy (linear plus

rotational)? [Note that the rotational inertia of a hollow

sphere is I = 2/3 MR2.]

Linear / Rotational Analogy

• θ, ω, α

• Torque: τ

• Rotational Inertia: I

• , ,

• Force:

• Mass: m

Linear Rotational Analogy

net

I

Kcm 1

2mv 2

• Newton’s

2nd law:

• Kinetic

energy:

K rot 1

2I 2

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Summary of some Different Types of Energy:

Kinetic Energy due to linear motion of centre of mass:

K = ½ mv2

Gravitational Potential Energy Ug = mgh

Spring Potential Energy: Us = ½ kx2

Rotational Kinetic Energy: Krot = ½ Iω2

Thermal Energy: Eth (often created by kinetic friction)

An object can possess any or all of the above.

One way of transferring energy to or out of an

object is work:

Work done by a constant force: W = Frcosθ

Last day I asked:

A hoop and a disk are both

released from rest at the top of

an incline. They both roll

without slipping. Which

reaches the bottom first? Shall

we vote?

A: hoop wins

B: disk wins

C: tie

Don’t forget: Nature is not a democracy!

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Let’s think about this. A solid disk is released from rest and

rolls without slipping down an incline. A box is released from

rest and slides down a frictionless incline of the same angle.

Which reaches the bottom first?

A: disk wins

B: box wins

C: tie

Compare and Contrast Soup Cans

• About same mass

• About same radius and

shape

• Thick paste, so when this

can is rolling, the contents

rotate along with the can as

one solid object, like a solid

cylinder

• Low viscosity liquid, so the

can itself rolls while the liquid

may just “slide” along.

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• Two soup cans begin at the top of

an incline, are released from rest,

and allowed to roll without

slipping down to the bottom.

Which will win?

Predict:

A. Cream of Mushroom will win

B. Chicken Broth will win

C. Both will reach the bottom at about the same time.

Soup Race

Initial:

Ug Klin Krot

Final:

Ug Klin Krot

Initial:

Ug Klin Krot

Final:

Ug Klin Krot

Cream of

Mushroom soup

must rotate, like

a solid disk.

Chicken broth

can slide down

without

rotating while

the can rotates

around it.

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From a student email this weekend

“I was wondering if content from the problem sets or

the practicals will be on term test 2?”

Harlow answer: Yes!

Tomorrow’s test is from 6:10 to 7:30 pm.

It will be based on the Uncertainties Reading plus

Wolfson Chapters 5-9 and Sections 10.1-10.3.

This includes everything covered in Classes 7-16,

Practicals 3-7, and MasteringPhysics Problem

Sets 3-7.

Tips for the 80 minute Test

• No phones / ipods etc allowed. You

will need a calculator, and a watch

could be handy as well!

• Time Management:

– Skim over the entire test from front

to back before you begin. Look for

problems that you have confidence

to solve first.

– If you start a problem but can’t

finish it, leave it, make a mark on

the edge of the paper beside it,

and come back to it after you have

solved all the easy problems.

• Bring your T-card or other photo ID,

as we will be collecting signatures

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Term Test 2 Info• The room you will write in is based on the first few letters

of your last name. You must attend the correct room, or

you will not be allowed to write the test (please note that

the assignment is different from that of Test 1):

• A – I: EX100

• J – R: EX200

• S – Tr: EX300

• Ts – Xi: EX310

• Xu – Z: EX320

• Alternate sitting students will receive a separate email by

Oct.13 letting you know the room and time.

• EX is the Exam Centre at

255 McCaul Street

Term Test 2 Info• Please be sure to bring your T-Card, as invigilators will be

collecting signatures and checking your photo-ID.

• Allowed aids are:

– A calculator with no communication ability

(programmable calculators and graphing calculators

are okay).

– A single hand-written aid-sheet prepared by the

student, no larger than 8.5”x11”, written on both sides.

– A hard-copy English translation dictionary.

– A ruler.

• If you wish to see the first page of the test as well as the

“Possibly Helpful Information”, I have posted it at

http://www.physics.utoronto.ca/~jharlow/teaching/phy131f

15/test2FirstPage.pdf .

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• The most important life-long resource in a class like

this is other students.

• Wednesday morning’s preclass quiz is optional (not for

marks!) You will be asked:

• Please tell me something interesting or notable about

the person who sits beside you during lectures. (Note:

this question is not required for marks. Please nothing

embarrassing!)

Remember:

Examples:

1. What is the acceleration of a slipping

object down a ramp inclined at angle θ?

[assume no friction]

2. What is the acceleration of a solid disk

rolling down a ramp inclined at angle θ?

[assume rolling without slipping]

3. What is the acceleration of a hoop

rolling down a ramp inclined at angle θ?

[assume rolling without slipping]

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1. What is the acceleration of a slipping

object down a ramp inclined at angle θ?

[assume no friction]

2. What is the acceleration of a solid disk

rolling down a ramp inclined at angle θ?

[assume rolling without slipping]

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2. What is the acceleration of a solid disk

rolling down a ramp inclined at angle θ?

[assume rolling without slipping]

3. What is the acceleration of a hoop rolling

down a ramp inclined at angle θ?

[assume rolling without slipping]

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Before Class 18 on Wednesday

The pre-class quiz is due by 10:00am and it is optional.

The reading is all of Chapter 11 on Rotational Vectors and

Angular Momentum.

Something to think about: When a figure-skater starts a spin

and brings in his arms, he spins even faster. Why?

Term test tomorrow from 6:10 to 7:30 pm.

The room you will write in is based on the first one or two

letters of your last name:

A – I: EX100

J – R: EX200

S – Tr: EX300

Ts – Xi: EX310

Xu – Z: EX320