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You may have had to find angular measure in math class. Angular measure is the angle measured between to lines that are not parallel. This can be considered to be a rotational displacement. If you  are spinning a rope over your head, the rope goes around and around. During a certain amount of time say a change in time of /\t, the rope would have moved around Ødegrees.

 Ø = s/r

where s is the arclength a specific point along the rope traveled and r is the radius of the circle that point traveled.

Angular Velocity

This is much like velocity, as its name says. It is the change of angular measure over a certain time interval.

 w = Ø t

Angular Acceleration

Do you see a trend here? Angular acceleration is acceleration no matter what word you put before it. It is the change in angular velocity  per unit time. This is the only angular measurement that is a vector.

 = w = wr2 t

Moment of Inertia

 I = mr2

The moment of inertia is the proportionality constant between net torque and angular acceleration.

[I] = kg·m2

 = Ia

Problem: A performer is a trying to impress the audience by twirling four sticks with different weight masses simultaneously. If he twirls it such that the two in each hand are spinning around as shown in the picture below, what is the moment of inertia of the baton system in each hand?

Torque

Torque is the ability of a force to rotate an object around an axis. The object is fixed at a point (which can be called the "origin" or pivot point) and then it spins around the axis given sufficient force exerted at the right places. Think of a monkey wrench as the mechanic is trying to screw in a loose screw tight. The wrench is fixed at the point where the screw is. The mechanic exerts a force to turn the wrench around the screw.

Terms

• pivot
• lever
• fulcrum

 Torque = = Fd

where d is the distance of the lever arm from the fulcrum, and F is the force applied at a point not at the axis of rotation.

Problem: Chubby and Skinny, who are walking in the park, spot a free see-saw and suddenly feel the urge to go sit on it. Skinny is dismayed by the fact that his see-saw partner overweighs him by...well 3 times his weight. So he suggests that Chubby move closer to the middle. Chubby inches up and plops down 2 meters away from the middle. How far away is Skinny sitting if Chubby weighs 270 kg?

Rotational Energy

Rotational Kinetic Energy

KE = (1/2)mv2, remember?

Well if we want to find the rotational kinetic energy, let's visualize what happens to the KE of an object as it is spun around.

Take a point in the object, and we'll call that A. A has a mass of m and a tangential velocity of v. All the particles in the object are spinning around an origin O. If r is the distance of point A to O (r is the radius of the circle formed by rotating point A around O), then the KE of point A is (1/2)mv2.

What about the KE of the entire object then? Well, if each point of mass has a certain KE, we must add up all these individual points

Since this is rotation, v = wr. Substitute that in for v to express the kinetic energy of rotation in rotational variables.

And from the equation for I, we find that:

 KErotational = (1/2)Iw2

Rotational Work

Angular Momentum

Well, now that we have established that t=Ia, we can analyze this too. What is a? Angular acceleration of course...but what

 L = Iw

[Angular Momentum] = [L] = kgm2/s

Conservation of Angular Momentum

You notice that figure skaters can execute great spins really fast? Well, it's all a part of physics.When the skater is spinning slowly her arms are out wide. Then as "she" pulls in her arms the spin gets faster and faster. Why is that? The skater has angular momentum when "she" is spinning.

 Li = Lf

You may encounter it in, most commonly, these two forms:

 (mvr)i = (mvr)f

 (Iw)i = (Iw)f

This is conservation of angular momentum.

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Key Themes
• Angular Measure
• Impulse
• Conservation of Linear Momentum
• Collisions
• Practice Questions

Java Interactions and Demonstrations

• none