Goal: To understand angular motionsObjectives:

1) To learn about Rotational Inertia2) To learn about Torque

3) To understand Angular Momentum4) To understand the Conservation of

Angular Momentum5) To understand the Affects on Earth due

to the conservation of angular momentum

Rotational Inertia

• If you want to know how something will accelerate linearly you need to know the force and mass.

• For circular acceleration the equivalent of the mass is called Rotational Inertia.

• Newton’s First law also applies here.• Something in rotation stays there unless

you act upon it.

Equations…

• For a very small in size object traveling in a circle the inertia for the object is:

• Inertia = mass * radius * radius• Where radius is the radius of the circle it is

moving in.

• For any not small object the inertia depends on how much of the mass is far from the point you are rotating around.

• The more mass further out the higher the inertia (the harder it is to spin something).

Getting into Inertia Shape

• A solid ball: I = 2/5 m r * r• A solid cylinder: I = ½ m r * r• A meter stick from an end: I = 1/3 m L * L• (L = length of stick)• A meter stick rotating around its center:

I = 1/12 m L * L

• A hoop spinning around its center:I = m r * r

• A hoop spinning on its side:I = ½ m r * r

Torque

• Now that we know about rotational mass we can examine rotational force!

• First of all lets see rotational acceleration:• Rotational acceleration = change in rotational velocity /

time

• Torque = force * distance from rotation pt• T = F * R

• Torque = Inertia * rotational acceleration• T = I * α

The torque challenge!

• A 30 kg kid sits on one end of a seesaw at a distance of 2.4 m from the center.

• A bigger kid, 60 kg, thinking for some reason that if he gets closer to the center that he can push more weight around get 0.7 m from the center.

• Which kid has more torque?

• Who will end up in the air?

Spin the wheel

• A wheel with an Inertia of 0.12 kg m2 is given a force of 18 N at a distance of 0.4 m from the center.

• A) What is the torque being applied to the wheel?

• B) What will the angular acceleration of the wheel be?

Another example

• A rod of very small mass (small enough to ignore) is attached to the side of a building. The length of the rod is 3.2 m

• At the end of the rod is tied a 2 kg mass.• At the center of the rod a string is tied which also

attaches to the building.• A) What is the torque that the 2 kg mass produces.• B) What is the torque that the string must produce in

order to keep the rod from rotating?• C) What is the vertical tension of the string?• D) If the string is strung to make a 30 degree angle with

the rod then what is the magnitude of the tension of the string?

2nd half: Angular Momentum

• Angular momentum = Inertia * Angular velocity

• (just like normal momentum = mv)

• L = I * ω

Quick sample

• A disc with an inertia of 0.4 kg m2 is spinning with an angular velocity of 12 rad/sec.

• What is the angular momentum of the spinning disc?

And in case you are wondering…

• Yes, angular kinetic energy = ½ Inertia * angular velocity * angular velocity

• But back to Momentum:• Angular mom = Inertia * Angular velocity

• And remember that:• Angular velocity = velocity / radius• Inertia = mass * radius * radius

• So, therefore, • Angular Momentum (L) = mass * velocity * radius

Conservation of angular momentum

• Just like with normal momentum, angular momentum is conserved!

• What does this mean?• Well, if you rotate, you stay rotating with

constant angular momentum.

• If you spin around the earth, you stay spinning!• So, Ltotal-before = Ltotal-after

Two discs

• Two discs each have an inertia of 0.4 kg m2.

• Initially the first disc is at rest while the 2nd disc is spinning at 6 rad/sec.

• A) What is the total angular momentum?• B) If the first disc is set on the 2nd disc so

that they will eventually rotate at the same angular speed what will the angular speed be? Hint, total angular momentum…

Hadley circulation• http://ess.geology.ufl.edu/ess/Notes/AtmosphericCirculation/atmoscell_big.jpeg

As air moves North or

South, it moves E/W

because of the spin

of the earth.

Going up in

Latitude means

you have less

rotational Energy

(smaller radius).

Therefore, to

conserve energy,

the air moves

westward.

Hurricanes

Coriolis Effect• Affects hurricanes/typhoons.• In southern hemisphere they spin backwards• Has to do with spin of earth and conservation of

angular momentum• HOWEVER, this does NOT affect toilets• The force is just too small to do anything

• Instead a given toilet/sink is designed to flush a certain way and can be designed to flush either direction

Conclusion

• Well, we have learned everything we could possibly want to know about angular motions.

• We see that once you get the inertia – or the rotational equivalent to mass, that all the equations for rotations are the same as for non rotations.

• Angular momentum is conserved, and this affects our weather – but no it does NOT affect our toilets!