Rotational Inertia and Torque in Physics
L-11 (M-10) Rotational Inertia and
Conservation of rotational momentum
1
•
Why does a wheel keep spinning?
•
Spinning ice skater
•
Why is a bicycle stable when it is
moving, but falls over when it stops?
•
A spinning wheel is difficult to turn.Video
Rotational inertia
symbol I
•
Rotational inertia
is a parameter that is
used to quantify how much torque it takes
to get a particular object rotating
•
it depends not only on the mass of the
object, but where the mass is relative to
the axis of rotation
•
the rotational inertia is bigger, if more
mass is located
farther
from the axis.
2
Rotational inertia and torque
•
To start an object spinning, a
torque must be applied to it
•
The amount of torque required
depends on the
rotational
inertia
(I)
of the object
•
The rotational inertia (I)
depends on the mass of the
object, its shape,
and on how
the mass is distributed.
I is
larger if more mass is further
from the center.
•
The larger the rotation inertia,
the more torque that is required
to make an object spin
W
=
m
g
T
Torque = T
R
3
rotational inertia examples
4
small rotational inertia
large rotational inertia
How fast does it spin?
•
For spinning or rotational motion, the
rotational inertia of an object plays the
same role as ordinary mass for simple
motion
•
For a given amount of torque applied to
an object, its rotational inertia determines
its rotational acceleration
the smaller
the rotational inertia, the bigger the
rotational acceleration
5
Big rotational
inertia
Small rotational
inertia
Same torque, different rotational inertia
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Rolling down the incline
Which one
reaches the
bottom first, the
solid disk or the
hoop? They
have the same
mass and
diameter.
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Speed of rotation
•
For motion in a straight line velocity is simply how far you
travel in a certain time (meters per second)
•
How do we quantify how fast an object rotates?
•
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•
for example, the rotational speed of the earth spinning on it
axis is 1 revolution per day or 1 revolution per 24 hours; the
rotational speed of the earth around the sun is 1 revolution
per year.
•
Another way to quantify rotational velocity is by the angular
displacement of the object in degrees per second
8
Ordinary (linear) speed vs. rotational speed
•
the rod is rotating around
the circle in the
counterclockwise
direction
•
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However, the red point in
the middle has only half
the linear speed as the
blue point on the end.
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9
Ice Capades
Skaters farther from center must skate faster
10
Hurricanes
The winds are higher as you get farther from
the eye of a hurricane.
11
Rotational (angular) momentum
•
Rotational, or angular momentum is a
measure of the amount of rotation an
object has, taking into account its
mass,
shape and speed.
•
It is a fundamental law of nature that the
total rotational (angular) momentum of a
system is constant.
12
Rotational (angular) momentum
•
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Rotational momentum depends on
–
the
mass
of the spinning object
–
where
the mass is located
–
how
fast
it is spinning
•
If the rotational inertia is larger, the rotational
momentum is larger
•
If the rotational velocity is larger,
the
rotational momentum is larger
13
Conservation of rotational momentum
•
If no outside torques disturb a spinning
object, it
rotational momentum is conserved
•
The rotating masses on the rod form a
system
and keep spinning until the friction
in the bearing slows brings it to rest.
•
Without friction in the axle, the system
would keep spinning indefinitely.
•
Note that the total
linear
momentum is zero.
14
Rotational momentum
•
The rotational momentum of a spinning object
depends on both its rotational inertia and its
rotational velocity (how fast it is spinning)
•
If either the rotational inertia or rotational velocity
changes, the other parameter must also change to
keep the rotational momentum constant
•
if the rotational inertia changes, the rotational
velocity must also change
•
If the rotational inertia increases, then the rotational
velocity must decrease
•
if the rotational inertia decreases, then the
rotational velocity must increases
15
Conservation of Rotational momentum
demonstrations
•
spinning ice skater
•
divers
•
Hobermann sphere
•
bicycle wheel
•
top
•
gyroscope
16
Big rotational
inertia
Small rotational
inertia
You can change your rotational inertia
17
Spinning faster or slower
•
When your arms are extended you have a
big rotational inertia
•
When you pull your arms in you make your
rotational inertia smaller
•
If you were spinning with your arms out,
when you pull your arms in you will spin
faster to keep your rotational momentum
constant
•
This works in figure skating and diving
18
Divers use rotational momentum
conservation to spin faster
•
the diver starts spinning
when she jumps off the
board
•
Her CG follows the path of
a projectile
•
when she pulls her arms
and legs in she makes her
rotational inertia smaller
•
this makes her spin faster!
19
Example
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Her final rotational velocity is 2 rev/sec.
20
Tornadoes (Cyclones)
•
Technical term is
mesocyclone
•
Intense updrafts stretch the
mesocyclone vertically
•
As it is stretched upward it
gets increasingly narrower
•
As it gets narrower, its rotation
speed increases
•
This is similar to the ice skater
who pulls hers arms in
21
Spinning wheel defies gravity!
•
An object that can
rotate about any axis is
called a gyroscope
•
Once it starts spinning
its axle wants to keep
spinning in the same
direction.
•
It resists forces that try
to change the direction
of its spin axis
22
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Falling off the stool!
http://www.youtube.com/watch?v=V3UsrfHa4MQ
23
The girl is holding
a spinning wheel
while sitting on a
stool that can
rotate. As she
rotates the wheel,
She also rotates!
Rotational inertia symbolizes how difficult it is to change the rotation of an object. It depends on mass distribution from the axis, shape, and mass. Torque needed to start rotation is proportional to rotational inertia. Smaller rotational inertia leads to faster rotational acceleration. Examples and illustrations further explain the concept's impact on spinning objects and stability in motion.
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Presentation Transcript
L-11 (M-10) Rotational Inertia and Conservation of rotational momentum Why does a wheel keep spinning? Spinning ice skater Video Why is a bicycle stable when it is moving, but falls over when it stops? A spinning wheel is difficult to turn. 1
Rotational inertia symbol I Rotational inertia is a parameter that is used to quantify how much torque it takes to get a particular object rotating it depends not only on the mass of the object, but where the mass is relative to the axis of rotation the rotational inertia is bigger, if more mass is located farther from the axis. 2
Rotational inertia and torque To start an object spinning, a torque must be applied to it The amount of torque required depends on the rotational inertia (I) of the object The rotational inertia (I) depends on the mass of the object, its shape, and on how the mass is distributed. I is larger if more mass is further from the center. The larger the rotation inertia, the more torque that is required to make an object spin R T M W= mg Torque = T R 3
rotational inertia examples small rotational inertia large rotational inertia m m m m M M M 4
How fast does it spin? For spinning or rotational motion, the rotational inertia of an object plays the same role as ordinary mass for simple motion For a given amount of torque applied to an object, its rotational inertia determines its rotational acceleration the smaller the rotational inertia, the bigger the rotational acceleration 5
Same torque, different rotational inertia Big rotational inertia Small rotational inertia spins fast spins slow 6
Rolling down the incline Which one reaches the bottom first, the solid disk or the hoop? They have the same mass and diameter. The solid disk gets to the bottom faster because it has a smaller rotational inertia 7
Speed of rotation For motion in a straight line velocity is simply how far you travel in a certain time (meters per second) How do we quantify how fast an object rotates? We use a parameter called rotational velocity, simply the number of revolutions per minute for example -- the number of times something spins say in a second or minute (rpm s- revs per min) for example, the rotational speed of the earth spinning on it axis is 1 revolution per day or 1 revolution per 24 hours; the rotational speed of the earth around the sun is 1 revolution per year. Another way to quantify rotational velocity is by the angular displacement of the object in degrees per second 8
Ordinary (linear) speed vs. rotational speed the rod is rotating around the circle in the counterclockwise direction ALL points on the rod have the SAME rotational speed every point moves through the same angle in the same time However, the red point in the middle has only half the linear speed as the blue point on the end. every point on the green line moves through the same angle 9
Ice Capades Skaters farther from center must skate faster 10
Hurricanes The winds are higher as you get farther from the eye of a hurricane. 11
Rotational (angular) momentum Rotational, or angular momentum is a measure of the amount of rotation an object has, taking into account its mass, shape and speed. It is a fundamental law of nature that the total rotational (angular) momentum of a system is constant. 12
Rotational (angular) momentum A spinning object has rotational momentum Rotational momentum depends on the mass of the spinning object where the mass is located how fast it is spinning If the rotational inertia is larger, the rotational momentum is larger If the rotational velocity is larger, the rotational momentum is larger 13
Conservation of rotational momentum If no outside torques disturb a spinning object, it rotational momentum is conserved The rotating masses on the rod form a system and keep spinning until the friction in the bearing slows brings it to rest. Without friction in the axle, the system would keep spinning indefinitely. Note that the total linear momentum is zero. 14
Rotational momentum The rotational momentum of a spinning object depends on both its rotational inertia and its rotational velocity (how fast it is spinning) If either the rotational inertia or rotational velocity changes, the other parameter must also change to keep the rotational momentum constant if the rotational inertia changes, the rotational velocity must also change If the rotational inertia increases, then the rotational velocity must decrease if the rotational inertia decreases, then the rotational velocity must increases 15
Conservation of Rotational momentum demonstrations spinning ice skater divers Hobermann sphere bicycle wheel top gyroscope Slow spin Fast spin 16
You can change your rotational inertia Big rotational inertia Small rotational inertia 17
Spinning faster or slower When your arms are extended you have a big rotational inertia When you pull your arms in you make your rotational inertia smaller If you were spinning with your arms out, when you pull your arms in you will spin faster to keep your rotational momentum constant This works in figure skating and diving 18
Divers use rotational momentum conservation to spin faster the diver starts spinning when she jumps off the board Her CG follows the path of a projectile when she pulls her arms and legs in she makes her rotational inertia smaller this makes her spin faster! 19
Example Question: A figure skater with her arms outstretched spins at the rate of 1 revolution per sec. By pulling her arms and legs in, she reduces her rotational inertia to one-half its value when her arms and legs were outstretched. What is her final rotational velocity? Solution: Her angular momentum is conserved. If her rotational inertia is reduced by a factor of 2, her rotational velocity must increase by a factor of 2. Her final rotational velocity is 2 rev/sec. 20
Tornadoes (Cyclones) Technical term is mesocyclone Intense updrafts stretch the mesocyclone vertically As it is stretched upward it gets increasingly narrower As it gets narrower, its rotation speed increases This is similar to the ice skater who pulls hers arms in 21
Spinning wheel defies gravity! An object that can rotate about any axis is called a gyroscope Once it starts spinning its axle wants to keep spinning in the same direction. It resists forces that try to change the direction of its spin axis spinning wheel 22
Falling off the stool! The girl is holding a spinning wheel while sitting on a stool that can rotate. As she rotates the wheel, She also rotates! L R L R 23 http://www.youtube.com/watch?v=V3UsrfHa4MQ
