Goal: To understand angular motions

Objectives:1) To learn about Circular Motions2) To learn about Rotational Inertia

3) To learn about Torque4) To examine Center of Mass

5) To learn about what causes Stability6) To understand the difference between Centripetal

Force vs Centrifugal force7) To understand Angular Momentum

8) To understand the Conservation of Angular Momentum

9) To understand the Affects on Earth due to the conservation of angular momentum

Circular Motion

• Previously we examined speed and velocity.

• However these were movements in a straight line.

• Sometimes motions are not straight, but circular.

Around and around

• If you rotate in a circle there will be a rate you rotate at.

• That is, you will move some angle every second.• w = angular velocity = change in angle / time• Units of w are radians/second or

degrees/second

• If you want a linear speed, the conversion is:• V = radius * angular velocity (in radians /

second)

Lets do an example.

• You are 0.5 m from the center of a merry-go-round.

• If you go around the merry-go-round once every 3.6 seconds (hint, how many degrees in a circle) then what is your angular velocity in degrees/second.

• There are 2 pi radians per circle. What is your angular velocity in radians per second?

• What is your linear velocity in meters per second?

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

A ball of mass 1.2 kg

• And a radius of 0.2 m.

• What is its Inertia if it is solid?

• If an ant of mass 0.01 kg is on the edge of the ball what is the inertia of the ant?

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

• Torque = Inertia * rotational acceleration

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?

Center of Mass

• Objects that are not tied down or held on an end will rotate around their center of mass.

• The center of mass is the average position of mass for an object.

• Note that for a weirdly shaped object, the center of mass can actually occur in a place where there is no mass.

• Where is the center of mass for a hollow soccer ball?

Sit down, stand up

• If I crouch down. What happens to my center of mass?

• When is this a good thing?

• When is this a bad thing?

• What happens to my center of mass if I lean forward?

Stability

• If I lean over at what point will I fall over?

Stability

• If I lean over at what point will I fall over?

• If my center of weight in the horizontal direction is at a point that is no longer supported by my base (such as my feet) then I fall over.

• Why do I fall over?

To support

• You just need to make sure your center of mass is supported.

Centripetal vs Centrifugal force

• These two are very similar.

• Centripetal force is a force that pulls you to the center.

• Gravity is an example here.

• When you are in circular motion, centrifugal force will try to push you out, and cancels out the centripetal force.

Equation

• Centrifugal force = mass * velocity * velocity / radius

• A spacecraft is in orbit around the earth at a distance of 6.5 * 106 m and a velocity of 8.2 * 103 m/s.

• A) If the mass of the spacecraft is 3000 kg then what is the centrifugal force on the spacecraft?

• B) How does that force compare to the gravitational force on the spacecraft (yes use mg here)?

Angular Momentum

• Did you notice that when I was talking about angular velocity, acceleration, and force, I left out momentum.

• Well, no longer.

• Just like with the other values,

• Angular momentum = Inertia * Angular velocity

• (just live normal momentum = mv)

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!

Ice skater

• An ice skater puts our their hands and spins.• Ang mom = Inertia * angular velocity.

• The skater then pulls their hands towards their body.

• A) What happens to the Inertia (remember that inertia is greater when mass is further from the rotational point)?

• B) Knowing that Angular momentum is conserved what must happen to the angular velocity?

Diver

• A diver with a height of 2 meters and a mass of 80 kg dives off a diving board.

• Initially he rotates around his center at an angular velocity of 3 radians/second.

• A) If his moment of inertia = 1/12 m * L * L then what is his moment of inertia and what is his angular momentum?

• B) He tucks into a ball with a radius of 0.5 m. Now the moment of Inertia = 2/5 m * r *r then what is his moment of Inertia?

• C) Knowing that angular momentum is conserved, what is his new angular velocity?

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

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!