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Physical Science: Help and Review19 chapters  248 lessons
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Angular momentum is the rotational equivalent to linear momentum. Both concepts deal with how quickly something is moving and how difficult it is to change that speed. However, linear momentum had only two variables: mass and speed. Angular momentum starts out with a very similar equation, so it may initially look to be about the same complexity. As we'll see, it gets more complicated and involves more variables than linear momentum.
Angular momentum is the product of an object's moment of inertia and its angular velocity. Both quantities must be about the same axis, the rotation line.
In the above equation, L represents angular momentum. The moment of inertia is represented by I, and the angular speed is omega. Let's see how this equation works.
The term 'moment of inertia' may sound confusing, but I'll bet you already know the basic concepts involved in calculating it. Let's say you have two rods of wood. They both weigh the same amount and have the same diameter, but one of them is 1 foot long and the other is 10 feet long. Which one will you be able to make spin faster if you throw it overhand towards a target? Does it take more energy to spin a ball around your head if it is on a 2 foot rope or a 5 foot rope? What if you double the mass of the ball?
You knew the smaller rod and shorter rope were going to be easier to spin, didn't you? So you're already on your way to calculating the moment of inertia for an object. You may be used to the word 'moment' referring to time, but in math and physics, moment refers to torque or twisting. So the moment of inertia refers to how difficult an object is to twist around a particular axis. The moment of inertia of any object is determined by three factors: shape, mass and the axis of rotation.
Here's a chart explaining the moment of inertia for different shapes:
Shape  Axis of Rotation  Moment of Inertia  

sphere  center of sphere 


point mass  distance r from point mass 


solid cylinder  axis of symmetry 


rod (length L)  center (not axis of symmetry) 


rod (length L)  one end 

The angular speed of an object is how quickly it is rotating about the axis of rotation. You can calculate the angular speed of an object if you know the linear speed of any point on it by using the equation omega = v / r. Omega is the angular speed, v is the linear velocity, and r is the distance from the axis of rotation.
Let's look at a few examples to help us understand how to calculate angular momentum.
Example 1  Point Mass
Ellen is spinning in a circle while attached to a rope with a 3 kg ball at the other end. The ball is traveling with a linear velocity equal to 10 m/s. Calculate the angular momentum of this ball.
The first thing to do here, as in many angular momentum problems, is to calculate the moment of inertia. This is an example of a point mass; in this situation, it doesn't matter what the shape of the weight is  only its mass is important. The calculations would be exactly the same if it were a cube, rectangular block or any irregular shape attached by the rope to Ellen.
I = (3 kg)(2 m)(2 m) = 12 kgm^2.
Now the angular speed:
Omega = v / r = (10 m/s) / (2 m) = 5s
Now with both values, we can determine the angular momentum.
L = Iw = (12 kgm^2)(5s) = 60 kgm^2/s
Example 2  Spinning Sphere
Matt works at a bowling alley, and during slow times he spins bowling balls on the counter top. What is the angular momentum of a 5 kg ball with a radius of 10 cm and spinning at 5 radians per second about an axis through its center?
Again, the first step is to calculate the moment of inertia using the table above.
I = 2(5 kg)(.1 m)(.1 m) / 5 = .02 kgm^2
Then we use that value in our angular momentum equation.
L = Iw = (.02 kgm^2)(5s) = 0.1 kgm^2/s
This amount is significantly less than in the example with Ellen and the weight on a rope  even though the angular speed is exactly the same and the ball is heavier this time. This is because it is much easier to spin a bowling ball about its own axis than it is to spin a weight on a much longer rope.
Done and done!
Example 3  Rod About Its End
Duanne the gorilla is spinning a 3 mlong stick (rod) over his head by grabbing one end. It has a mass of 12 kg and its far end is moving with a linear velocity of 21 m/s. What is the angular momentum of the stick?
Moment of inertia:
I = (12 kg)(3 m)(3 m) / 3 = 36 kgm^2
For this one, we also need to translate linear velocity to angular speed:
w = (21 m/s) / (3m) = 7 radians/s
The last step is to use both of those values in our angular momentum equation:
L = Iw = (36 kgm^2)(7 radians/s) = 252 kgm^2/s
This is even more angular momentum than our previous examples. The lesson here  gorillas are very strong!
Angular momentum is the product of an object's moment of inertia and its angular speed around the same axis, given by the equation:
The moment of inertia depends on the object's mass, shape, and the axis of rotation. The angular speed of an object is how quickly it's rotating about the axis of rotation.
Angular momentum  the rotational equivalent to linear momentum; the product of an object's moment of inertia and its angular velocity
Moment of inertia  how difficult an object is to twist around a particular axis
Angular speed  how quickly an object is rotating about the axis of rotation
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Physical Science: Help and Review19 chapters  248 lessons