Upload
vothu
View
215
Download
1
Embed Size (px)
Citation preview
1
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.
2
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
3
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!
4
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.
5
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:
6
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.
7
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
8
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
9
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.
10
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
11
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!
12
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.
13
• 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.
14
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
15
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 .
16
• 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]
17
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]
18
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]
19
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