Physics 211 Lecture 16, Slide 1

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Physics 211 Lecture 16
Today's Concepts Rolling, Kinetic energy,
Angular Acceleration
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Exam Results
Average 77.5
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You gave us a lot of comments about the HW.
Do well on the clicker questions and we will 1)
Extend HW 10 deadline until Friday 2) Show you
how to do a few of the problems
Dear Tim and Mats, This stuff is making me
rethink how my world works, and it sort of rocks.
Please continue to make this happen. Sincerely,
Jim P.S. Mats, your dog rocks.
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Lectures 17 (Wednesday) and 18 (next Monday) will
focus on working statics problems. No new
concepts are involved so you wont have
pre-lectures for these (though you will still
have pre-flights).
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Rolling Motion

• Objects of different I rolling down an inclined
  plane

v 0 ?? 0 K 0
?K - ?U Mgh
R
M
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Rolling

• If there is no slipping

v
?
v
v
Where v ?R
In the lab reference frame
In the CM reference frame
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Rolling
v
hoop c 1 disk c 1/2 sphere c
2/5 etc...
Use v ?R and I cMR2 .
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Act

• A bowling ball (uniform solid sphere) rolls along
  the floor without slipping. What is the ratio of
  its rotational kinetic energy to its
  translational kinetic energy?

A) B) C)
Recall that for a solid sphere about
an axis through its CM
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Prelecture
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) Block B) Ball C) Both reach the same height.
v
w
v
66 got this right
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The block slides without friction and the ball
rolls without slipping. Which one makes it
furthest up the ramp?
v
A) Block B) Ball C) Same
w
v
A) For the ball, some of the kinetic energy is
used to produce rotation.
B) The ball has rotational kinetic energy as
well, so Ktotal for the ball is more. Hence it
will convert to greater potential energy --gt
greater height.
C) Using the conservation of energy, both have
the same initial kinetic energy, so they will
therefore have the same potential energy, and
reach the same height.
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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?
Prelecture

• A) Cylinder
• B) Hoop
• C) Both reach the bottom at the same time

42 got this right
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Which one reaches the bottom first?

• A) Cylinder
• B) Hoop
• C) Both reach the bottom at the same time

A) Both objects have the same initial potential
energy, but since the hoop has a larger
rotational inertia, more energy will be converted
into rotational kinetic energy than the cylinder,
which will mean the cylinder will have more
translational kinetic energy and reach the bottom
first.
B) The hoop has a greater moment of inertia, and
therefore has a greater rotational kinetic energy.
C) both has same mass and are at the same height
so they must have the same kinetic energy when
they reach the bottom
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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?
Prelecture

• A) Small cylinder
• B) Large cylinder
• C) Both reach the bottom at the same time

41 got this right
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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 cylinder
• B) Large cylinder
• C) Both reach the bottom at the same time

A) The small cylinder reaches the bottom first
because it has a lower moment of inertia than the
large heavy cylinder.
B) The large cylinder starts with more
gravitational energy so by the end it will have a
higher speed.
C) They both have the same moment of inertia and
mass cancels out from both sides of the
conservation of energy equation.
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A small light cylinder and a large heavy cylinder
are released at the same time and slide down the
ramp without friction. Which one reaches the
bottom first?
Act

• A) Small cylinder
• B) Large cylinder
• C) Both reach the bottom at the same time

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Act

• Suppose a cylinder (radius R, mass M) is used as
  a pulley. Two masses (m1 gt m2) are attached to
  either end of a string that hangs over the
  pulley, and when the system is released it moves
  as shown. The string does not slip on the pulley.
• Compare the magnitudes of the acceleration
• of the two masses
• a1 gt a2
• a1 a2
• a1 lt a2

M
?
R
T2
T1
a2
m2
m1
a1
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Act

• Suppose a cylinder (radius R, mass M) is used as
  a pulley. Two masses (m1 gt m2) are attached to
  either end of a string that hangs over the
  pulley, and when the system is released it moves
  as shown. The string does not slip on the pulley.
• How is the angular acceleration of the
  wheelrelated to the linear acceleration of the
  masses?
• a Ra
• a a/R
• a R/a

M
?
R
T2
T1
a
m2
m1
a
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Act

• Suppose a cylinder (radius R, mass M) is used as
  a pulley. Two masses (m1 gt m2) are attached to
  either end of a string that hangs over the
  pulley, and when the system is released it moves
  as shown. The string does not slip on the pulley.
• Compare the tension in the string on either side
  of the pulley
• T1 gt T2
• T1 T2
• T1 lt T2

M
?
R
T2
T1
a2
m2
m1
a1
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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.

y
x
M
For the hanging masses use F ma -m1g T1
-m1a -m2g T2 m2a
?
R
T2
T1
a
m2
m1
a
m2g
m1g
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• We have three equations and three unknowns (T1,
  T2, a). Solve for a.
• -m1g T1 -m1a (1)
• -m2g T2 m2a (2)
• T1 - T2 (3)

y
x
M
?
R
T2
T1
a
m2
m1
a
m2g
m1g
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Two identical disks rotate about fixed axles.
Identical masses are attached to strings wrapped
around the axle and the outside rim of the two
disks, respectively. The masses are released
from rest at the same height. Just before each
mass hits the ground, which disk has more
rotational kinetic energy?

• A) Disk I
• B) Disk II
• C) Same

Ground
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Disk With Hole



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Pivoting Stick
cm
F - Mg Macm Mw2R
F Mg Mw2R
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Loop the loop
h
R
If it just makes it around, N0