classical mechanics lecture 16 today’s concepts: a) rolling kinetic energy b) angular acceleration...
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Classical Mechanics Lecture 16
Today’s Concepts:a) Rolling Kinetic Energyb) Angular Acceleration
Physics 211 Lecture 16, Slide 1
Schedule
Mechanics Lecture 14, Slide 2
One unit per lecture!
I will rely on you watching and understanding pre-lecture videos!!!!
Lectures will only contain summary, homework problems, clicker questions, Example exam problems….
Midterm 3 Wed Dec 11
Main Points
Mechanics Lecture 16, Slide 3
Main Points
Mechanics Lecture 16, Slide 4
Rolling without slipping Rv /
Total Kinetic Energy = Translational Kinetic Energy + Rotational Kinetic Energy
Energy Conservation
Rotational Kinetic Energy
Mechanics Lecture 16, Slide 5
H
0KEMgHPE
0PErottrans KKKE
hgv
MvhMgKU
hMgU
tot
7
10
10
70 2
Rolling without slipping
22
2
1;
2
1 IKMvK rottrans
Rv /
2
22
22
10
7
5
1
5
2
2
1
2
1
MvKKK
MvR
vMRIK
rottranstot
rot
Newton’s Second Law
Mg
f a
Acceleration of Rolling Ball
Mechanics Lecture 16, Slide 6
xmafmg sin
Newton’s 2nd Law for rotations
CMCMCMnet I ,
fRCMnet ,
CMCMCMnet IfR ,
CMCMI
fR
R
axCM Rolling without slipping
R
a
MR
fR x2
52
xMaf5
2
xx mamamg 5
2sin
xmamg5
7sin
sin7
5gax
Rolling Motion
Objects of different I rolling down an inclined plane:
v 0
0
K 0
K U M g hR
M
h
v = R
22
2
1
2
1MvIK
Mechanics Lecture 16, Slide 7
Rolling
If there is no slipping:
In the lab reference frame In the CM reference frame
v vv
Where v R
Mechanics Lecture 16, Slide 8
Rolling
Use v R and I cMR2 .
Doesn’t depend on M or R, just on c (the shape)
Hoop: c 1
Disk: c 1/2
Sphere: c 2/5
etc...
2222 12
1
2
1
2
1MvMvMRK c c
v
22
2
1
2
1MvIK
MghMv 212
1So: c c 1
12
ghv
Mechanics Lecture 16, Slide 9
Clicker QuestionA. B. C.
Mechanics Lecture 16, Slide 10
33% 33%33%
A) B) C)
A hula-hoop rolls along the floor without slipping. What is the ratio of its rotational kinetic energy to its translational kinetic energy?
Recall that I MR2 for a hoop about
an axis through its CM:
1trans
rot
K
K
4
3
trans
rot
K
K
2
1
trans
rot
K
K
1
2121
2
1)(
2
1
2
12
1
2
2
2222
2
Mv
Mv
K
K
MvR
vMRIK
MvK
trans
rot
rot
trans
A block and a ball have the same mass and move with the same initial velocity across a floor and then encounter identical ramps. The block slides without friction and the ball rolls without slipping. Which one makes it furthest up the ramp?
A) BlockB) BallC) Both reach the same height.
CheckPoint
v
v
Mechanics Lecture 16, Slide 11
The block slides without friction and the ball rolls without slipping. Which one makes it furthest up the ramp?
A) BlockB) BallC) Same
v
v
B) The ball has more total kinetic energy since it also has rotational kinetic energy. Therefore, it makes it higher up the ramp.
Mechanics Lecture 16, Slide 12
CheckPoint
Rolling vs Sliding
Mechanics Lecture 16, Slide 13
g
vh
MvhMgKU
hMgU
MvKKK
MvR
vMRIK
ball
tot
rottranstot
rot
2
2
2
22
22
10
7
10
70
10
7
5
1
5
2
2
1
2
1
Rolling Ball Sliding Block
g
vh
MvhMgKU
hMgU
MvKKK
K
MvK
block
tot
rottranstot
rot
trans
2
2
2
2
2
1
2
10
2
1
02
1
5
7
105
107
2
2
gvgv
h
h
block
ball Ball goes 40% higher!
Rolling without slipping
22
2
1;
2
1 IKMvK
KKK
rottrans
rottranstot
Rv /
CheckPoint
A cylinder and a hoop have the same mass and radius. They are released at the same time and roll down a ramp without slipping. Which one reaches the bottom first?
A) CylinderB) HoopC) Both reach the bottom
at the same time
Mechanics Lecture 16, Slide 14
A) CylinderB) HoopC) Both reach the bottom
at the same time
A) same PE but the hoop has a larger rotational inertia so more energy will turn into rotational kinetic energy, thus cylinder reaches it first.
Which one reaches the bottom first?
Mechanics Lecture 16, Slide 15
CheckPoint
A small light cylinder and a large heavy cylinder are released at the same time and roll down a ramp without slipping. Which one reaches the bottom first?
A) Small cylinderB) Large cylinderC) Both reach the bottom
at the same time
Mechanics Lecture 16, Slide 16
A small light cylinder and a large heavy cylinder are released at the same time and roll down a ramp without slipping. Which one reaches the bottom first?
A) Small cylinderB) Large cylinderC) Both reach the bottom
at the same time
C) The mass is canceled out in the velocity equation and they are the same shape so they move at the same speed. Therefore, they reach the bottom at the same time.
Mechanics Lecture 16, Slide 17
CheckPoint
Mechanics Lecture 16, Slide 18
vf RM
II
I
Rf
2
52
RM
RgM
R
g
2
5
Mechanics Lecture 16, Slide 19
vf RM
a
M
Mg g
M
Fa MaF
v
a RM
t
Once vR it rollswithout slipping
t
v0R Rt
0v v at t
Mechanics Lecture 16, Slide 20
t
v
v0
atvv 0
Rv Rtatv 0
ga
R
g
2
5
tg
gtv2
50
tg
v2
70
07
2v
gt
v
a RM
Plug in a and t found in parts 2) & 3)
Mechanics Lecture 16, Slide 21
ga
07
2v
gt
20 2
1attvx
atvv 0
v
a RM
We can try this…
Interesting aside: how v is related to v0 :
Mechanics Lecture 16, Slide 22
ga
07
2v
gt
atvv 0
g
vgvv
7
2)( 0
0
00 7
2vvv
Doesn’t depend on 07
5vv
0714.0 vv
v
f RM
Mechanics Lecture 16, Slide 23
2
2
1MvKtran
2
2
1 IKrot 2
5
1Mv
22
5
2
2
1
R
vMR
Atwood's Machine with Massive Pulley:
A pair of masses are hung over a massive disk-shaped pulley as shown.
Find the acceleration of the blocks.
For the hanging masses use F ma
m1g T1 m1a
m2g T2 m2a
For the pulley use IR
aI
T1R T2R MRaR
aI
2
1
(Since for a disk)2
2
1MRI
Mechanics Lecture 16, Slide 24
y
x
m2gm1g
aT1
a
T2
R
M
m1m2
We have three equations and three unknowns (T1, T2, a).
Solve for a.
m1g T1 m1a (1)
m2g T2 m2a (2)
T1T2 (3)Ma2
1
gMmm
mma
221
21
Mechanics Lecture 16, Slide 25
y
x
m2gm1g
aT1
a
T2
R
M
m1m2
Atwood's Machine with Massive Pulley:
Three Masses
Mechanics Lecture 16, Slide 26
amTT
aRmR
aRm
R
aIIRTT
diskspherehoop
diskdiskdisk
diskdiskdisk
diskdiskdiskdiskspherehoopnetdisk
2
1
2
1
2
1 2,
amfamTamfT spherespherespherespherespherespheresphere 5
7
amf
aRmR
aIIRf
spheresphere
spherespheresphere
spherespherespherespherespherenetsphere
5
2
5
2,
)( agmTamTgm hoophoophoophoophoop
spherehoopdisk
hoop
hoopspherehoopdisk
diskspherehoop
mmm
gma
gmmmma
amamagm
57
21
5
7
2
12
1
5
7)(
sphereT
hoopT
sphereT
hoopT
spheref
Three Masses
Mechanics Lecture 16, Slide 27
spherehoopdisk
hoopsphere
mmm
gmaa
5
7
2
1
diskspherehoopdisk
hoop
diskdisk
diskdisk
Rmmm
gm
R
a
R
a
5
7
2
1
spherespherehoopdisk
hoop
spheresphere
spheresphere
Rmmm
gm
R
a
R
a
5
7
2
1
Three Masses
Mechanics Lecture 16, Slide 28
amfamTamfT spherespherespherespherespherespheresphere 5
7
)
57
21
(
)(
spherehoopdisk
hoophoophoop
hoophoophoophoophoop
mmm
gmgmT
agmTamTgm
a
yt
aty
2
2
1 2
Three Masses
Mechanics Lecture 16, Slide 29
yaa
yaatv
a
yt
aty
22
2
2
1 2
tR
at
R
at
spheresphere
spherespheresphere
Three Masses 2
Mechanics Lecture 16, Slide 30
gmmm
mma
amgmT
amTgm
amTT
2
2
1
321
21
222
111
321
Three Masses 2
Mechanics Lecture 16, Slide 31
pulleypulley R
a
agmT 11
agmT 22
2
2
1aty
tpulleypulley
atv
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