In Exercises 17-58, solve the given problems. Most DVD...

In Exercises $17-58,$ solve the given problems. Most DVD players use constant linear velocity. This means that the angular velocity of the disk continually changes, but the linear velocity of the point being read by the laser remains the same. If a DVD rotates at 1590 r/min for a point $2.25 \mathrm{cm}$ from the center, find the number of revolutions per minute for a point $5.51 \mathrm{cm}$ from the center.

In Exercises $17-58,$ solve the given problems.
Most DVD players use constant linear velocity. This means that the angular velocity of the disk continually changes, but the linear velocity of the point being read by the laser remains the same. If a DVD rotates at 1590 r/min for a point $2.25 \mathrm{cm}$ from the center, find the number of revolutions per minute for a point $5.51 \mathrm{cm}$ from the center.

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Key Concepts

Rotational Motion and Unit Conversion

In problems involving rotational systems, it is common to convert between different units of rotational speed, such as revolutions per minute. Understanding how to manipulate units and apply the constant linear velocity condition is essential in solving for unknown rotational parameters.

Constant Linear Velocity

In systems where the linear velocity remains constant, every point along the rotating object's surface moves with the same tangential speed regardless of its distance from the center. This concept is key to understanding how different points on a rotating disk behave when the disk is designed to maintain a uniform read speed, for example.

Relationship Between Linear and Angular Velocity

The fundamental relationship in rotational motion is given by the equation v = r?, where v is the tangential (linear) velocity, r is the radius, and ? is the angular velocity. This concept allows one to determine how a change in radius affects angular velocity when the linear speed is held constant.

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