1) One thing to remember: ω is
the angular (rotational) speed. The quantity tells you how fast
an object is spinning. The units are in radians/s.
So if ω = 0.5
rad/s , this means that the object sweeps 0.5 radians or 30 degrees per
second/ As shown previously, there is a relationship between the angular speed ω and the tangential speed V, also called the orbital speed : V = R ω. R is the radius (usually in meters if V is in m/s)
A) 2 cars are spinning at the same rate. They covers the same angle
(30
degrees) in one second. . The yellow one goes from C to D and the green
one goes from A to B.
They both sweeps 30 degrees in 1 second. So they both have the same __________ speed. (orbital ? angular?)
ω = ______ degrees/sec = _______ radians / sec
(remember ? pi is for 180 degrees)
Obviously arc AB > arc CD so, during the same time, the ____ car (green or yellow)
covers more distance than the ____ car. So the green ccar moves
_______. In other words, the orbital speed of the green car is ________
than the orbital speed of the yellow car. Even though they have the same angular speed !
B) If ω
= 30 degrees/sec, R1 = 100m and R2 = 200m, find the speed V1
(orbital speed of the yellow car ) and V2 (orbital speed of the red
car). Of course you all have in mind the relationship: V = R ω
V1 = __________m/s and V2 = ____________ m/s
hint: you need to convert
degrees/sec to radians/s, solve the proportion: 180degrees is for 3.14
radians so 30 degrees is for ...
C) conclusion: In the example studied, if you increase the
radius (moving farther from the center), you
increase the tangential speed or orbital speed because you
cover more distance. This is also true for an old record or a CD
spinning. Any point on the same radius will have the same angular speed
but not the same orbital speed if the distance from the center differs.
If the radius of a CD in a computer is 0.06m and the disk turns
at constant angular speed of 31.4 rad/s , what is the tangential speed
of a microbe riding on the disc's rim ?
D) A revolving door turns at an angular speed of 1.75rad/s. If a woman
passes through the door at a distance of 1 m from the center of the
door, what is the woman's linear speed ?
E) if an object has an angular
speed of 6.28 rad/s (about), it will complete _____ degrees every
second or ____ revolution (1 tour)
2) In the previous example, the car were spinning at a constant rate.
In that case, the same angle is swept every second. If the car
accelerates along the track, they will increases their speed (tangential and angular) every
second. They will have an angular or rotational acceleration α .
For example, if they increases their spinning rate by 0.5 rad/s every second, α = 0.5/s/s.
A)Can you guess the relationship between α the angular acceleration and aT the tangential acceleration (or orbital) ? aT = ___α
B) A spinning ride at a carnival has an angular
acceleration of 0.5 rad/s/s. Find the tangential acceleration
of a rider who sits 6.5 m from the center.
C) A dog sits 1.5 m from the center of gravity of a merry go round, If
the dog undergoes a tangential acceleration of 1.5m/s/s, what is the
angular acceleration of the merry go round ?
3) please read: During the first trimester we studied linear motion
(object going in a straight line, not rolling or spinning, acted upon
by forces). Newton's
second law says:
An object, of mass m (its inertia), acted upon by an
unbalanced force F, will accelerate with an acceleration a: F = m a.
F is the unbalanced force
a is the the linear acceleration (linear because the object moves in a straight line)
m is the linear inertia (or
mass in that case) .
Inertia is the resistance of an object to change
its motion. More inertia (more mass) and harder it will be to
start moving the object, to slow it down or to speed it up.
(first law: a object at rest, stays at rest, an object in motion, wants to stay in motion, same speed and same direction)
WE are now studying circular motion. REmember? to rotate an object (if
you need to open a door) , you need to apply twist called a _____________
(a force
is not enough, if the force is along the door, you are not going to
move it about the axis)
4) Read please. For a circular motion, Newton's second law (F = ma) becomes: Tr = I α
Tr :
is the torque that will produce the rotational or angular acceleration α(
if you keep pushing a merry go round, without friction, it will keep
going faster, sweeping larger angle every second). Units: N.m
I:
is the rotational inertia also called the angular inertia. THat is the
resistance of the object to its rotational motion. It depends not
only on the mass of the object but it also depends on the distribution
of the mass about the axis of rotation. Units: kg.m2
(I will explain below)
α: rotational
acceleration. rad/s/s
5) THe angular inertia (rotational inertia) I,
depends not only on the mass of the object (a heavier door is harder to
open than a light one) but it also depends on the distribution of the
mass about the axis. Do the following experiments to convince
yourself:
A) get a meter stick with 2 mass holders. Get 2 big masses (1kg). First
hang the 2 masses at both end of the meter stick (see below) and try to
rotate it holding it at its center. Bring the 2 masses closer to the
axis. Is it harder ? or easier ?
____________________________
When the masses are closer to the axis, the rotational inertial I (or angular inertia) is ____________________
because it is ________________ to rotate the system/
Conclusion: It is harder to get the system rotating when the mass is at the end of the rod than when it is nearer the axis.
B) Here is another experiment. Have someone standing on a stool.
The courageous candidate has to hold heavy weight in both hands. Is it
easier to rotate her/him when her/his hands are along his body or
stretch out ?
It is easier to rotate the system when the masses are ___________________ from the axis.
C) On an inclined plane, make a race between 2 cylinders. The first
cylinder has more mass close to the axis and the second one has more
masses on the edge. Which one wins the race ?
D) You can try a race between a loop and a solid cylinder and the cylinder wins !
(try for example with an empty can and a can filled with dog food, the filled should win)
So is it easier to push a merry go round when the kids st inside or outside ? _______________________________
E) The angular inertia depends not only on the mass of the rotating
object but also on the distribution of the mass around the axis. Here
some examples.
A ball of mass m attached
to a string rotating
I = m r2
|
A ring rotating:
I = m r2 |
A disk rotating:
I = 0.5 m r2
|
A sphere rotating
I = 0.4m r2 |
A rotating rod:
I = 0.33.. m r2= I = 1/3 m r2 |
Which system resist the most to acceleration ? _____________________
Which system resists less to a change in its state of motion? _____________________________
For
the same mass and radius , which object is easier to rotate : a
cylinder (mass m, radius r, axis in the middle) or a mass m attached to
a string of radius r and spinning ? ____________
A loop or a sphere ? _________ (same mass, same radius, axis of rotation in tne middle)
F) Surprisingly, if loops and cylinders (disks) race on a
inclined plane, the cylinder always wins,
no matter the mass is ! (only works for race on inclined plane)
watch that movie (u-tube)
conclusion: all the hoops roll ________, all the disks roll ___________ but the disks always __________ the race.
Watch the second part of the movie
All the spheres roll ________ and the spheres always _______ the disks !
G) In case of a race, the mass does not matter (only true for a race on
an inclined plane, no friction). Only the distribution of the mass
about the axis matters.
Can you show this interesting fact using the conservation of energy ?
hint: linear kinetic energy + rotational kinetic energy = potential energy
with linear KE = 0.5mV2, rotational KE = 0.5 I ω2, PE = mgh
, ω is the angular speed, the angle in radian covered in 1 second.
H) So which one will win such a race, a cube (not rolling so ω= 0) or a sphere ? _________________________
6) Please read:The last interesting thing about circular motion is the
conservation of angular momentum.
Remember the linear momentum from 10th
grade ? p = mV.
The linear momentum is a quantity that tells you how hard it is to stop an object moving in a straight line.
p: is the momentum, it is a vector. Unit= kg m/s
m : is the inertia of the object (mass in this case)
V is the linear
velocity of the object, (the object is going in a straight line, not spinning )
If an object spins or rolls, it has an angular momentum.
The quantity angular momentum tells you how hard it is to stop a spinning object.
Can you find a similar equation for the angular momentum L ? L = __________ (Unit = kg m2/s)
hint: use ω the angular speed (the angle swept in one second, units are radians) and I the angular inertia or rotational inertia.
REmember: p mV for a linear motion.
Unit for ω are rad/s. (we suppose the motion uniform, ω is constant)
I is the angular inertia. Unit = kg m2
7) A) please read: Remember last year ? If there is no external force acting on a system, the system conserved its linear momentum p = mV.
If a pluck of mass m slides on the ice, with a given direction and a
given speed, it will conserve its momentum. It will keep the same
velocity. This conservation of momentum was really convenient when
studying collision. If 2 plucks collide, we can predict in which
direction and at what speed they will go after the collision because: ptotal before = ptotal after.
(see 10th grade class)
Now, as long as there is no external torque applied to a spinning system, its angular momentum L is also conserved. So L1 = L2
L1 is the angular momentum of the system before.
L2 is the angular inertia after.
OR
I1ω1 = I2ω2
I1 is the inertia before and ω1 (Unit rad/s) is the angular or rotational speed before
I2 is the inertia after and ω2 is the angular speed after.
B) Using this principle can you explain why a skater will spin faster
if she brings her arms next to her body ? __________________ (hint: In this case we have I1ω1 = I2ω2 see picture )
_______________________________________________________
_______________________________________________________
C) Kepler found out that the planets have elliptical orbit around the
Sun/ They sweep equal area during the same amount of time. (Remember
the
lab). To do so, they need to spin faster when they get closer to the Sun
and slower when they are farther away. Can you explain this fact using
the conservation of angular
momentum ?
It
means, there is no external torque applied to the planets since they
conserve their angular momentum. What about the force of gravity that keep the solar system together?
Doesn't it apply a torque ? why ?
hint: torque = force x radius and the force has to by perpendicular to the radius.
D) You can experience the conservation of angular momentum using a stool ? explain
E)
E) Imagine a whirlpool forming as you are emptying a bath tub. The water
spins faster and faster as it gets closer to the drain, forming a
vortex. There is no external torque applied to the water. The water
tends to move toward the drain because of the pressure differential.
(there is an inward force, toward the center but no torque). Why is it
spinning faster ?
A
tornado is a whirlpool upside down. Made out of air instead of water.
The air moves up creating a zone of low pressure. Air moves to replace
it and spins at the same time but faster and faster. Why ?
The
shape of galaxies can also be explain by the conservation of angular
momentum. The particles pick up speed as they moves inward. What force
keeps the galaxy together ? _______________
The force is inward so the torque of the force is ________ so the angular momentum is _________.
source image: http://www.rondianetyler.com/tornado.aspx
This image shows the devastating 1870 fire of Chicago.
Look at the flames. What is the connection with what we are studying ?
8) A) REad:The angular momentum is a vector. So not only the magntitude but also the direction have to be conserved if no external torque is applied. This can result in strange phenomena like the
precession of a top, a cat always landing on his feet or a bicycle wheel
processing about a string. Watch the U-tube movies
the bicycle wheel the stool and wheel the cat falling
Try it ! You sit on a stool still. The stool is not spinning. The
instructor gives you a spinning bicycle wheel. The angular momentum is
a vector and is up. Let's call the vector 1 up. (magnitude = 1 and direction = up)
See picture on the left. Flip the wheel upside down. There is no external torque applied to the system wheel + you + stool. (you do apply a torque by you are part of the system)
Now the angular momentum is 1 down. To keep its original value 1 up, the stool has to spin 2 up.
Get it ? 2 up + 1 down = 1 up
So by flipping the wheel you make the stool spins by itself. Physics is really super extra COOL !
B)Can you knkow understand the cartoon posted on the website. Why is the girl spinning ?
hint: The system girl + EArth need to conserved their total momentum. There is no external torque.
It
seems that the recent tsunami has changes the angular momentum of the
Earth and therefore its angular speed slightly because a big fault
formed.
prepare index cards with:
1) Angular momentum =L = I
ω I is the angular inertia ( kg m2), ω is
the
angular (or rotational) speed (rad/s or rev/s)
2) conservation of momentum L1 = L2
or I1ω1= I2ω2 or V1 R2 = V2 R2
L is the angular momentum ( kg m2/s)
3) Newton's 2nd law = Tr = I α Tr
is the net torque (N.m) acting on a rotating
object (a torque can be negative)
α is the angular acceleration (rad/s2)
4) V = R ω
R is the
radius from the axis.
5) aT = R α
aT (m/s2) is the tangential acceleration
9) An ice skater has a rotational inertia of I1 = 1.2 kgm2 when her arms are extended and a rotational inertia of I2 = 0.5 kgm2
when her arms are pulled in close to her body. If she goes into a spin
with her arm extended and has an initial rotational velocity of 1
rev/s, what is her rotational velocity when she pulls her arms in, close
to her body ? (you don't need to convert rev/s to rad/s. )
hint: use the conservation of momentum I1
ω1 = I2 ω2
10)See index card in pink, above. Make your own. try without hints, pleaeaease ..
A merry go round is rotating at a rate of 10 rev/min
A) express this rotational speed in rev/s
B) express this speed in rad/s
hint: 1 revolution = 2 pi radians and 1 minute = 60 seconds, of course
11) A net torque of 60 Nm is applied to a disk with rotational inertia of 12 kgm2.
What is the rotational acceleration of the disk ?
hint: use Tr = I α
12) A wheel with a rotational inertia of 4.5 kg m2 accelerates at a rate of 3.0 rad/s2. What net torque is needed to produce this acceleration ? (hint: use the same equation as for 10)
13) A torque of 50N.m producing a counterclockwise rotation (the torque
is positive, Tr2 = 50N.m) ) is applied to a wheel about its axle. A
frictional torque of 10 N.m (the torque is negative, Tr1 = -10 N.m)
acts about the axle.
A) What is the net torque about the axle ?
hint: net torque = sum of the 2 torques,
B) If the wheel is observed to accelerate at the rate of 2 rad/s2 under the influence of these torques, what is the rotational inertia of the wheel ?
net torque = inertia x acceleration (rad/s2)
14) Consider a ball of mass m spinning at the end of a string of radius R
(like the demonstration in class)
A) what is the expression of its angular inertia ? I =
____________ (see above 4E, the
table)
B) If the ball is going at V = 2.5 m/s (V is the tangential or orbital
speed) when the radius of its path is R= 40 cm , how fast
will it be going when the string is pulled enough to reduce the radius
to 15 cm ? ( convert cm to m first, try not to convert, do you get the same thing? why?)
hint: First find the angular speed ω using V = R ω, then use the conservation of momentum I1 ω1 = I2 ω2, solve for ω2
15) A 40kg boy is on a merry go round that spins at one
revolution every 6s. If he is 3.5 m from the center of rotation, what
is his angular momentum?
hint: first find ω in radians. 1 revolution = 2pi, the use L = I ω, with I = m r2
16) A catapult propels a m= 0.150kg stone in the air. Assume that
the length of the catapult arm is r= 0.350m.and that its moment of
inertia is negligible. If the stone leaves the catapult with an
acceleration of α = 100 m/s2, what is the torque exerted on the stone?
hint: use Tr = I α, I = mr2
( advanced or extra credits )
17) A 65kg person is spinning on a merry go round that has a mass
of 115kg and a radius of 2.0m. She walks from the edge of the merry go
round toward the center, If the angular speed of the merry go round is
initially 2.0 rad/s, what is its angular speed when the student reaches
a point 0.5m from the center ?
hint: Treat the student as a point mass (I = mr2) and treat the merry
go round as a disk (I = 0.5 mr2). The total inertia = inertia of
student + inertia of merry.
18) advanced
A 0.11kg mouse rides on the edge
of a Lazy Susan that has a mass of 1.3kg and a radius of 0.25m. If the
lazy sysan begins with an angular speed of 3 rad/s , what is its
angular speed after the mouse walks from the edge to a point 0.15m from
the center ?
hint: you should get 3.2 rad/s
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