On a 12.0-cm-diameter audio compact disc (CD), digital bits...

On a 12.0-cm-diameter audio compact disc (CD), digital bits of information are encoded sequentially along an outward spiraling path. The spiral starts at radius $R_1 = 2.5 cm$ and winds its way out to radius $R_2 = 5.8 cm$. To read the digital information, a CD player rotates the CD so that the player's readout laser scans along the spiral's sequence of bits at a constant linear speed of 1.25 m/s. Thus the player must accurately adjust the rotational frequency $f$ of the CD as the laser moves outward. Determine the values for $f$ (in units of rpm) when the laser is located at $R_1$ and when it is at $R_2$.

On a 12.0-cm-diameter audio compact disc (CD), digital bits of information are encoded sequentially along an outward spiraling path. The spiral starts at radius $R_1 = 2.5 cm$ and winds its way out to radius $R_2 = 5.8 cm$. To read the digital information, a CD player rotates the CD so that the player's readout laser scans along the spiral's sequence of bits at a constant linear speed of 1.25 m/s. Thus the player must accurately adjust the rotational frequency $f$ of the CD as the laser moves outward.
Determine the values for $f$ (in units of rpm) when the laser is located at $R_1$ and when it is at $R_2$.

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Relationship between Linear and Angular Speed

In rotational motion, the linear (or tangential) speed at a point on a circle is directly related to the angular speed and the radius of the circle through the formula v = ?r. This relationship allows one to determine the necessary angular speed to achieve a desired linear speed at various radii on a rotating disc.

Conversion of Angular Speed Units

Angular speed is often calculated in radians per second, but in many applications it is necessary to convert this value into revolutions per minute (rpm). Since one complete revolution corresponds to 2? radians and there are 60 seconds in a minute, the conversion factor allows one to translate the angular speed into a more practical unit for applications like rotating media.

Circular Geometry and Rotational Kinematics

Understanding the basic properties of circles, such as radius and circumference, is crucial in rotational systems. The distance traveled in one complete rotation (the circumference) depends directly on the radius, influencing the angular speed required to maintain a specific linear speed along different paths of a spiral.

Transcript

Please give Ace some feedback