PHY131H1F - Class 16

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PHY131H1F - Class 16

Today is the last day of material that is testable for Midterm Test 2:

10.1 Angular Velocity and Acceleration

10.2 Torque

10.3 Rotational Inertia and the Analog of Newton’s Law

Quick Quiz

The time derivative of ω is α. What are the S.I. units of α ?

A. m/s2

B. N

C. N/m

D. rad

E. rad /s2

d

dt

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Class 16 Preclass Quiz on MasteringPhysics

This was due this morning at 8:00am

68% of students got: A girl and a boy are riding on a merry-

go-round that is turning at a constant rate. The girl is near

the outer edge, and the boy is closer to the center. Both the

girl and the boy have the same nonzero angular

displacement.

91% of students got: Two equal forces are applied

perpendicular to a door. The first force is applied at the

midpoint of the door; the second force is applied at the

doorknob. The force applied at the doorknob exerts the

greater torque.

Class 16 Preclass Quiz on MasteringPhysics

76% of students got: A constant net torque is applied to a

rotating object. The object will rotate with constant angular

acceleration.

89% of students got: A single particle has a mass, m, and it

is at a distance, r, away from the origin. The rotational

inertia of this particle about the origin is mr2.

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Class 16 Preclass Quiz Student Comments

“The relationship between the Torque and the

Rotational Inertia takes time to think about..”

Harlow comment: It is the same as the

relationship between Force and Mass. The angular

acceleration is directly proportional to the torque

(force) and inversely proportional to the rotational

inertia (mass).

“Would we be given an integral table if a question

requires integration?”

Harlow answer: Yes. Except for polynomial. You

should know, or have on your aid sheet that

𝑥𝑛𝑑𝑥 =1

𝑛+1𝑥𝑛+1


“Are shadows made up of atoms?”

Harlow answer: No. A shadow is the lack of light in

an area. Even light is made of photons, which are

massless particles (ie pure energy).

“Do you believe in aliens????”

Harlow answer: I do not believe there is any

evidence to indicate that life exists in the universe

outside the earth, yet. However, I do believe and

hope that evidence will come forward in my lifetime.

Somewhere in the universe there is probably

intelligent life, and they probably sent out radio

signals into space, just as we do now. If we

continue to fund SETI, it should succeed.

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“Should we know how to derive the rotational inertia's found

in Table 10.2? Or can we just memorize them?”

Harlow answer: I will put needed rotational inertia equations

from Table 10.2 on the test or final, if needed. You can also

write them on your aid sheet if you like.


“I am I remember in an early lecture you said that as a child,

you wanted to dig a hole through the centre of the Earth to

the other side, and you got about 1 metre deep. I just

learned that although there is still no hole all the way

through, the world's deepest hole is located on the Kola

peninsula in northwest Russia near Finland, and they got

about 1.23x10^4 metres deeper than you did, but it took

them 24 years!”

Harlow comment: Okay, that’s 12 km. And it’s 6370 km to

the centre, and another 6370 km “up” to the other side.

Even the Russians have a long way to go!

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“A friend dropped this physics course, It feels like

Rose leaving Jack sinking into the water (in this

case I am Jack).”

Get to know someone!• There is an optional pre-class quiz due on the morning of

Wednesday Nov. 18 by 10:00am.

• I will post some comments in class that day, just for fun.

• On Monday’s class, sit CLOSER (if you can!)

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Angular Position

𝑠 = 𝑟𝜃

NOTE: This equation

only works if θ is

measured in radians.

Angular Velocity

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In-Class Discussion Question :

A carnival has a Ferris wheel

where some seats are located

halfway between the center

and the outside rim. Compared

with the seats on the outside

rim, the inner cars have

A. Smaller angular speed and greater tangential speed

B. Greater angular speed and smaller tangential speed

C. The same angular speed and smaller tangential speed

D. Smaller angular speed and the same tangential speed

E. The same angular speed and the same tangential speed

Rigid Body Rotation

Angular velocity is

The units of α are rad/s2.

All points on a rotating rigid

body have the same ω and

the same α.

The units of ω are rad/s. If the rotation is speeding up or slowing down, its angular acceleration is

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• Arc length: 𝑠 = 𝜃𝑟

• Tangential velocity: 𝑣𝑡 = 𝜔𝑟

• Tangential acceleration: 𝑎𝑡 = 𝛼𝑟

The fan blade is speeding up. What are the signs of and ?

A. is positive and is positive.

B. is positive and is negative.

C. is negative and is positive.

D. is negative and is negative.

Historical Convention:

Define positive angular displacement to

be counter-clockwise.

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The fan blade is slowing down. What are the signs of and ?

A. is positive and is positive.

B. is positive and is negative.

C. is negative and is positive.

D. is negative and is negative.

Historical Convention:

Define positive angular displacement to

be counter-clockwise.

Rotational Kinematics

• θ is angular position.

The S.I. Unit is

radians, where 2π

radians = 360°.

• Angular velocity:

• Angular acceleration:

• s (or x or y) specifies

position.

• Velocity:

• Acceleration:

Linear Rotational Analogy

vx d

dtx

ax d

dtvx

d

dt

d

dt

vy d

dty

ay d

dtvy

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Radians are the Magical Unit!

• Radians appear and

disappear as they

please in your

equations!!!

• They are the only

unit that is allowed

to do this!

• Example: 𝑣𝑡 = 𝜔𝑟

Rotational Kinematics

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Last day I asked at the end of class:

• Why is a door easier to open when the

handle is far from the hinge, and more

difficult to open when the handle is in the

middle?

• ANSWER:

• Torque is the rotational analog of force:

• Force causes things to accelerate along a line.

• Torque causes things to have angular acceleration.

• Torque = Force × Lever Arm

• Lever arm is the distance between where you apply the

force and the hinge or pivot point.

• Putting the handle further from the hinge increases your

lever arm, therefore it increases your torque for the same

applied force.

Rotational Dynamics

• θ

• ω

• α

• Torque: τ

• Rotational Inertia: I

• x

• vx

• ax

• Force: Fx

• Mass: m

Linear Rotational Analogy

Newton’s Second Law:

ax (Fnet )x

m

net

I

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Example

The engine in a small airplane is

specified to have a torque of 60.0

N m. This engine drives a

propeller whose rotational inertia

is 13.3 kg m2. On start-up, how

long does it take the propeller to

reach 200 rpm?

TorqueConsider the common experience of pushing open a door. Shown is a top view of a door hinged on the left. Four pushing forces are shown, all of equal strength. Which of these will be most effective at opening the door?

A. F1

B. F2

C. F3

D. F4

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TorqueConsider the common experience of pushing open a door. Shown is a top view of a door hinged on the left. Four pushing forces are shown, all of equal strength. F1 is most effective at opening the door.

The ability of a force to cause a rotation depends on three factors:1. the magnitude F of the force.2. the distance r from the point of application to the pivot.3. the angle at which the force is applied.

TorqueThe effectiveness of a force at causing a rotation is called

torque. Torque is the rotational equivalent of force. We say that a

torque is exerted about the pivot point.

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TorqueMathematically, we define

torque τ (Greek tau) as

SI units of torque are N m.

English units are foot-pounds.

Luis uses a 20 cm long wrench to

turn a nut. The wrench handle is

tilted 30° above the horizontal, and

Luis pulls straight down on the end

with a force of 100 N. How much

torque does Luis exert on the nut?

Example

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Rotational inertia is

A. the rotational equivalent of mass.

B. the point at which all forces appear to act.

C. the time at which inertia occurs.

D. an alternative term for lever arm.

Quick quiz..

Rotational Inertia

Depends upon:

• mass of object.

• distribution of mass

around axis of rotation.

– The greater the distance

between an object’s mass

concentration and the axis,

the greater the rotational

inertia.

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Consider a body made of N particles, each of mass mi,

where i = 1 to N. Each particle is located a distance ri

from the axis of rotation. For this body made of a

countable number of particles, the rotational inertia is:

The units of rotational inertia are kg m2. An object’s rotational inertia depends on the axis of rotation.

For a continuous distribution of mass (uncountably high number of particles), you must use an integral to compute rotational inertia:

𝐼 = 𝑟2𝑑𝑚

Rotational Inertia

Which dumbbell has the larger rotational inertia about

the midpoint of the rod? The connecting rod is

massless.

m

R

m/2

2R

m m/2A B

A. Dumbbell A.

B. Dumbbell B

C. Their rotational inertias are the same.

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Rotational Inertias of Simple Objects


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The Parallel-Axis Theorem

• Suppose you know the

rotational inertia of an

object when it rotates

about axis through center

of mass: Icm

• You can find the rotational

inertia when it is rotating

about another axis, which

is a distance d away from

the center of mass:

I = Icm + Md 2

Term Test 2 Info

• The second term test in PHY131H1F will be

written on Tuesday, Nov 17, 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 material was covered in Classes 7-16,

Practicals 3-7, and MasteringPhysics problem

sets 3-7.

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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 .

http://www.physics.utoronto.ca/~jharlow/teaching/phy131f15/test2FirstPage.pdf

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Before Class 17 Monday

Remember, there are Practicals this week from now until

Tuesday (including today!) - This week’s Practicals include

testable material for Test 2 on Tuesday evening!

For Monday please finish reading Chapter 10. Monday’s

material will not be testable on Test 2, but it will be on the final

exam.

Something to think about:

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?

Why?

PHY131H1F - Class 16

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