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Chapter 10: Problem 44

A compact disc (CD) player varies the rotation rate of the disc in order to keep the part of the disc from which information is being read moving at a constant linear speed of \(1.30 \mathrm{m} / \mathrm{s}\) Compare the rotation rates of a 12.0 -cm-diameter CD when information is being read (a) from its outer edge and (b) from a point \(3.75 \mathrm{cm}\) from the center. Give your answers in \(\mathrm{rad} / \mathrm{s}\) and rpm.

Short Answer

Expert verified
The rotational speed of the CD at the outer edge is approximately \(21.67 \, \mathrm{rad/s}\) or \(207 \, \mathrm{rpm}\), and at the point \(3.75 \mathrm{cm}\) away from the center, it is approximately \(34.67 \, \mathrm{rad/s}\) or \(330 \, \mathrm{rpm}\).

Step by step solution

01

Calculate the Angular Velocity at the Outer Edge

The radius of the CD is half of its diameter, so first calculate the radius of the entire disc, which is \(6.0 \mathrm{cm} = 0.06 \mathrm{m}\). Then, use the formula for linear speed and angular velocity: The linear speed (v) is equal to the radius (r) times the angular velocity (\(\omega\)), so rearrange the equation to solve for the angular speed:\[\omega = \frac{v}{r} = \frac{1.3 \mathrm{m/s}}{0.06 \mathrm{m}} \approx 21.67 \mathrm{rad/s}\]
02

Convert Rad/s to RPM for the Outer Edge

To convert from rad/s to revolutions per minute (rpm), use the conversion factors. Since 1 revolution is \(2\pi\) radians, the conversion factor from rad/s to rpm is \(\frac{60}{2\pi}\). \[\mathrm{rpm} = \omega \mathrm{(rad/s)} \times \frac{60}{2\pi} \approx 21.67 \times \frac{60}{2\pi} \approx 207 \mathrm{rpm}\]
03

Calculate the Angular Velocity at a Point 3.75 cm from the Center.

Now, repeat this analysis for a radius of \(3.75 \mathrm{cm} = 0.0375 \mathrm{m}\).\[\omega = \frac{v}{r} = \frac{1.3 \mathrm{m/s}}{0.0375 \mathrm{m}} \approx 34.67 \mathrm{rad/s}\]
04

Convert Rad/s to RPM for the Inner Point.

Again, convert the angular speed from rad/s to rpm:\[\mathrm{rpm}=\omega \mathrm{(rad/s)} \times \frac{60}{2\pi} \approx 34.67 \times \frac{60}{2\pi} \approx 330 \mathrm{rpm}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angular Velocity
Angular velocity is a measure of how quickly an object rotates or spins around an axis. In simple terms, it describes the speed of rotation. For example, when you are playing a CD, the angular velocity defines how fast the CD is turning in its player.
This is important because it ensures that the data on the CD is read at a consistent rate, which affects the quality and consistency of the playback.To calculate angular velocity, you use the formula:
  • \( \omega = \frac{v}{r} \)
  • where \( \omega \) is the angular velocity in radians per second (rad/s), \( v \) is the linear speed of the edge of the disc, and \( r \) is the radius of the circular path.
This concept is especially significant because it determines how long it takes for the CD to make one complete rotation. Changes in angular velocity can also affect the speed of reading or writing on a disc.
Linear Speed
Linear speed refers to how quickly something moves along a path. In the case of a CD player, it means the speed at which the information under the laser's focus travels.To maintain consistent playback and data reading, a CD player is designed to keep the linear speed constant across its surface. This ensures the laser reads the data accurately, no matter where on the disc it is.The formula to find linear speed, \( v \), is:
  • \( v = r \times \omega \)
  • where \( r \) is the radius of the path, and \( \omega \) is the angular velocity.
Because it's so important for the linear speed to remain constant, the CD player adjusts its angular velocity depending on whether the laser is reading near the center or near the edge of the disc.
Rotation Rate
The rotation rate refers to how many times an object, such as a CD, completes a full circle or rotation every second or minute. It's a crucial part of how CD players work. Rotation rate is often described using terms like revolutions per minute (rpm) or radians per second (rad/s). These measures help us understand how quickly the disc spins in the player. The rotation rate is adjusted by the CD player to keep the linear speed constant as data is read. When a CD player reads data closer to the center, it rotates faster (higher rotation rate) since the radius is smaller. Conversely, it rotates slower when reading near the outside, optimizing the reading speed and accuracy.
Conversion Unit rad/s to rpm
Understanding how to convert rad/s to rpm is essential when dealing with rotations and circular motion applications like CD players.Radian per second (rad/s) is a unit of angular velocity, while revolutions per minute (rpm) is a unit of rotation rate. To compare or convert between these, remember:
  • 1 revolution equals \( 2\pi \) radians.
  • To go from rad/s to rpm, multiply the angular velocity in rad/s by \( \frac{60}{2\pi} \). This accounts for the number of seconds in a minute and the radians in a full circle.
  • For example, if a CD spins at 21.67 rad/s, you multiply by \( \frac{60}{2\pi} \) to get approximately 207 rpm.
Converting these units helps us understand the speed of rotation in more practical or easy-to-visualize terms, making it easier to analyze or solve problems.

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Most popular questions from this chapter

You're asked to check the specifications for a wind turbine. The turbine produces a peak electric power of 1.50 MW while turning at its normal operating speed of 17.0 rpm. The rotational inertia of its rotating structure- three blades, shaft, gears, and electric generator-is \(2.65 \times 10^{7} \mathrm{kg} \cdot \mathrm{m}^{2} .\) Under peak conditions, the wind exerts a torque of \(896 \mathrm{kN} \cdot \mathrm{m}\) on the turbine blades. Starting from rest, the turbine is supposed to take less than 1 min to spin up to its 17 -rpm operating speed. The generator is supposed to be \(96 \%\) efficient at converting the mechanical energy imparted by the wind into electrical energy. During spin-up, the electric generator isn't producing power, and the only torque is due to the wind. Once the turbine reaches operating speed, the generator connects to the electric grid and produces a torque that cancels the wind's torque, so the turbine turns with constant angular speed. Does the turbine meet its specifications?

A solid sphere and a solid cube have the same mass, and the side of the cube is equal to the diameter of the sphere. The cube's rotation axis is perpendicular to two of its faces. Which has greater rotational inertia about an axis through the center of mass?

A car tune-up manual calls for tightening the spark plugs to a torque of \(35.0 \mathrm{N} \cdot \mathrm{m} .\) To achieve this torque, with what force must you pull on the end of a 24.0 -cm-long wrench if you pull (a) at a right angle to the wrench shaft and (b) at \(110^{\circ}\) to the wrench shaft?

A circular saw spins at 5800 rpm, and its electronic brake is supposed to stop it in less than 2 s. As a quality-control specialist, you're testing saws with a device that counts the number of blade revolutions. A particular saw turns 75 revolutions while stopping. Does it meet its specs?

A point on the rim of a rotating wheel has nonzero acceleration, since it's moving in a circular path. Does it necessarily follow that the wheel is undergoing angular acceleration?
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