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Conservation of Angular Momentum: Understanding CD-Gum Interaction Dynamics

This article explores the conservation of angular momentum through the example of a CD and a piece of gum. We delve into a scenario where a CD with a radius of 58.0 mm and mass of 15.2 g is spinning at 3750 RPM when a 0.500 g piece of gum sticks to it at half its radius. We calculate the changes in angular velocity as the system transitions from the initial state to a final state of angular speed following the conservation principles. This investigation highlights the relationship between mass, radius, and rotational energy loss in rotating systems.

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Conservation of Angular Momentum: Understanding CD-Gum Interaction Dynamics

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Presentation Transcript

  1. Conservation of Angular Momentum Important points to consider: • The units of angular velocity, , are rad/s. • Use Rev/min instead of rpm so you can see the units cancel.

  2. gum Radius The Problem: A compact disk, CD, has a mass of 15.2 g and a radius of 58.0 mm. The angular speed varies as it reads the outmost track versus the innermost track. Let’s say a song is playing which resides close to the middle of the CD and it is turning at 3750 RPM. Unfortunately, a piece of gum with a mass of 0.500 g falls onto, and sticks to, the CD at a point midway between its rim and center. Also, at this same instant, the CD loses power and is freely turning. Assuming no friction, calculate the final speed in RPM of the CD-gum combination.

  3. Treat the CD as a disk: Treat the gum as a point mass: The gum colliding with the CD is the “event” and:

  4. Let’s begin by finding a few values and doing some converting. This will make our expressions a little cleaner later on. ; Not rotating ½ of CD radius

  5. There are two objects, so there will be two terms for initial and two terms for final. 0 Now, because the CD and gum are stuck together, they have a common final speed:

  6. So, the speed of the CD decreases from 393 rad/s to 388 rad/s. The final speed corresponds to less rotational energy than the CD had originally. The “lost” energy shows up in the rotational energy of the gum.

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