Download Lecture slides with notes

Today: rotational versions of familiar things…
Mass/inertia (Moment of inertia)
Kinetic energy (Rotational KE)
Momentum (Angular momentum)
Impulse (τΔt = Δangular momentum)
Conservation of momentum (…of angular momentum)
…and conservation of energy
Extra Practice: 8.43, 8.49, 8.51, 8.55, 8.65
Today we’ll go at rapid pace through a few concepts that you’ve seen before, but now we’ll be applying them to rotational motion. If you had trouble in the last exam or in the past few weeks with any of these concepts now is really
the time to come meet with me, the TAs, or a tutor, or simply brush up, because you’ll be doing more or less the same types of problems now for rotational motion!
Today’s Main Ideas
Moment of inertia, I (kinda like mass)
Rotational momentum, L (kinda like p)
Rotational kinetic energy, KEr (kinda like KEt)
Reminder: Torque
τ = rF⊥
On Monday I was saying that torque can be thought of as the equivalent of Force in linear motion. I.e. you need a force to make something accelerate, or change its
angular velocity. And I mentioned that equally, you need a torque to make something rotate at a faster or slower angular velocity.
***moment of inertia ( I )***
Relates net torque ( Στ ) to angular acceleration (α).
Remember Newton’s
second law?
Rotational
version…
ΣF=ma
Στ = Iα
THINGS KEEP MOVING
at same v
UNLESS YOU APPLY A FORCE!
THINGS KEEP SPINNING
at same ω
UNLESS YOU APPLY A TORQUE!
Net torque related net force to acceleration by its mass.
Compare to Newton’s 1st law: The velocity of an object does not change (a=0) unless acted on by a force ΣF=ma ! Equivalently, an object will KEEP SPINNING or
ROTATING unless you apply a torque to it. This, based on the previous slide, means we have to apply a perpendicular force.
In this analogy, MOMENT OF INERTIA is the rotational equivalent of mass. But it’s not mass! So what is it?
Moment of Inertia
Depends not just on mass but also
how the mass is distributed!
I = Σmr2
Think of it as a measure of “rotational laziness”. It’s hard to rotate things with large moments of inertia, easy to rotate things with small rotational inertia.
Moment of Inertia
Depends not just on mass but also
how the mass is distributed!
I = Σmr2
Think of it as a measure of “rotational laziness”. It’s hard to rotate things with large moments of inertia, easy to rotate things with small rotational inertia.
Moment of Inertia
Think of it as “takes more effort to rotate”
(R1≠R2)
We are never going to make you determine something’s moment of inertia using the I=sigma mr2 formula unless it’s for discrete objects. For instance, the rotational
inertia of a person holding a hoop could be considered a rod and a hoop; in that case, you add the moments of inertia for the rod and the hoop. There’s something like
this on the homework tonight. Anyways, you’ll typically either calculate it or be given it.
If the mass and outer R are the same,
which one of these I is smallest?
A.
B.
C.
D. Two are
the same
ANSWER: B
(R1≠R2)
Q85
Remember that things with mass distributed at larger radii are harder to rotate! So the easiest thing to rotate is something with more mass at the center. CAN SEE THIS MATHEMATICALLY: A is largest. C is just over half of A, and B is exactly half of A. So B, a solid disk, has the smallest moment of inertia: it’s the easiest to
rotate. MASS FAR IN, MASS FURTHER OUT.
Race of the Geometrical Shapes
Which shape will win the race?
Note: since all of these shapes have the same mass, they all have the same
force that will act to get them rolling down the incline (equal torque).
All have the same mass and radius
I= mr2 1/2 mr2 2/5 mr2
Hoop Disk Sphere
τ = rF
Στ = Iα
They all have the same torque.
Thus, smaller I means a larger
angular acceleration, which is
also a larger linear acceleration
[HINT: forget the answer to the last one and think through this anew]
The thing with the smallest I will win because it’s easiest to get rolling. They all have the same FORCE OF GRAVITY ACTING DOWN THE RAMP, but that force of gravity
rolls the sphere most easily.
Look at what the angular acceleration will be. Torque over MOMENT OF INERTIA!. THEREFORE Smaller your I, the larger your alpha to get the same torque. You’ll spin
faster and get down the hill faster! So smaller I means larger net acceleration down the ramp!
HOOP: Lazy because big I. Disk: Less lazy, smaller I. Sphere: least lazy. Easy to get rolling.
All shapes below have the same mass and same radius
In order to maximize the moment of inertia (I),
the mass should be:
I= mr2 1/2 mr2 2/5 mr2
Hoop Disk Sphere
A. Concentrated at the edges
B. Evenly distributed
C. Concentrated at the center
D. Makes no difference
Q86
ANSWER: A
All shapes below have the same mass and same radius
In order to maximize the moment of inertia (I),
the mass should be:
I= mr2 1/2 mr2 2/5 mr2
Hoop Disk Sphere
A. Concentrated at the edges
B. Evenly distributed
C. Concentrated at the center
D. Makes no difference
Q86
ANSWER: A
Rotational Kinetic Energy
KEt = 12 mv2
KEr = ?
I’d like you to practice the symbol recognition of what they mean in rotational motion… So linear acceleration translating to an angular quantity, alpha, and mass
translating to its rotational equivalent, moment of inertia, etc. So following the symbolic analogy we’ve been doing, can you imagine what rotational kinetic energy should
be? Of course if you did the reading this should be particularly easy!
ANSWER: E
Remembering the similarities between the concepts can help your intuition and make these concepts feel less new.
Rotational Kinetic Energy
KEt = 12 mv2
KEr = ?
A. mv2
B. 0.5 mv2
C. 0.5 Iα2
D. 0.5 mω2
E. 0.5 Iω2
Q87
I’d like you to practice the symbol recognition of what they mean in rotational motion… So linear acceleration translating to an angular quantity, alpha, and mass
translating to its rotational equivalent, moment of inertia, etc. So following the symbolic analogy we’ve been doing, can you imagine what rotational kinetic energy should
be? Of course if you did the reading this should be particularly easy!
ANSWER: E
Remembering the similarities between the concepts can help your intuition and make these concepts feel less new.
It requires extra energy to rotate things!
A 1.20 kg disk with a radius of 10.0 cm rolls
without slipping. The linear speed of the disk is
v = 1.41 m/s.
(a) Find the translational kinetic energy.
(b) Find the rotational kinetic energy.
(c) Find the total kinetic energy.
Conservation of Energy
Work done by things like friction (= 0 if not slipping)
=
Sum of PE, KEr, KEt final - Sum of PE, KEr, KEt initial
I’m going to try something that I think will help you guys learn: since we’ve already covered conservation of energy, I want you you to try to puzzle through the application
of this concept to rotational motion yourself, to help you build intuition for these equations and how to use them. To do that, one of the homework problems tonight is
calculating the velocity of a ball rolling down a hill (#6). We’ll go over another example Friday I think in case you’re not getting it, but for now I will say just a few things to
set you up. IF IT’S ROLLING WITHOUT SLIPPING, DON’T WORRY ABOUT WORK LOST TO FRICTION. SET WORK TO ZERO. If it slips, YOU LOSE SOME ENERGY TO
FRICTION. WORK IS NOT ZERO. Just like in linear CoE.
Angular Momentum
Angular momentum
L = Iω
(Remember I = Σ mr2)
Units:
kg m2/sec
[Could also be written rad kg m2/sec]
Linear momentum depended on inertia/mass and velocity.
Angular momentum depends on rotational inertia and rotational velocity.
Reducing r makes ω greater to have same angular momentum.
Conservation of (Angular)
Momentum
Angular momentum
ΣLf = ΣLi
is conserved in an
isolated system!
Reducing r changes your moment of inertia! This makes ω greater to have same angular momentum. This is how conservation of angular momentum will often show up in
problems.
This shows up a lot in sports. Gotta respect these guys’ awesome control of their moments of inertia. In this example, you’re changing the MOMENT OF INERTIA to influence the
angular momentum.
Conservation of (Angular)
Momentum
If no unbalanced torque acts on a rotating system,
the angular momentum of that system is constant.
τ net ,system
ΔL
= ∑τ =
=0
Δt
Angular momentum:
Makes the World Go Round (literally!)
But you can also conserve angular momentum by changing ANGULAR VELOCITY instead of moment of inertia!
Let’s explore another implication of this… We saw earlier that the angular acceleration will change if we have a torque[see light board notes]. if we hold I constant, to
get a change in angular momentum, in other words a change in the rotation rate, you need a torque applied over some amount of time.
THIS SHOULD LOOK FAMILIAR: IT LOOKS A LOT LIKE IMPULSE! Let’s apply this. Does the Sun apply a torque? No, Fg is all along the radial direction (F parallel). THERE IS
NO PERPENDICULAR FORCE! Only inertia!
A Windmill
In a light wind, a windmill experiences a constant
torque of 255 N m.
If the windmill is initially at rest, what is its angular
momentum after 2.00 s?
τ=
ΔL
Δt
Notice that you did not need to know the moment
of inertia of the windmill to do this calculation.
Now let’s have a torque acting.
[See light board notes for solution]
Relations
Linear Motion
Mass m
Moment of Inertia I
Linear velocity v
Angular velocity ω
Translational KE
1
2
mv 2
Linear momentum p = mv
Δp
Fnet = ΣF =
Δt
(F=ma)
when m constant
Here it is all laid out.
Let’s practice.
Rotational Motion
Rotational KE
1
2
Iω 2
Angular momentum L = Iω
τ net = ∑τ =
Note: if I is constant,
ΔL
Δt
ΔL Iω − Iω o IΔω
=
=
= Iα
Δt
Δt
Δt
Extreme
fusion:
star
EXPLODES!
Core of
star
r = 2.3 x 108 m
Gravity
collapses core
Neutron
star!
r = 10 km
Before star dies, it rotates at ω0 = 2.4 x 10-6 rad/s.
What is the final spin rate of the neutron star?
At the end of a star’s life…This is asking for final angular velocity. Perhaps it’s a conservation of angular momentum problem! Let’s try that. The core actually collapses so
much that it can’t squish anymore because the neutrons are holding it together!
A WHOLE FREAKING STAR SPINNING AS FAST AS A DENTIST’S DRILL!
Quit your major and
do astronomy!
This actually happens a lot!
We know of about 2200 examples of this in the galaxy. Researchers at WVU are finding new examples all the time. If you think this sounds awesome, change your major
and come do research with us!
Quit your major and
do astronomy!
This actually happens a lot!
We know of about 2200 examples of this in the galaxy. Researchers at WVU are finding new examples all the time. If you think this sounds awesome, change your major
and come do research with us!