Chapter 8: Problem 3
A CD rotates at \(22.0 \mathrm{rad} / \mathrm{s}\). What is its angular speed in revolutions per minute \((\mathrm{rpm})\) ?
Short Answer
Expert verified
The angular speed is approximately 210.0 rpm.
Step by step solution
01
Understanding Angular Speed
The angular speed given is in radians per second (\( \text{rad/s} \)). We need to convert this into revolutions per minute (\( \text{rpm} \)). 1 revolution is equivalent to \( 2 \pi \) radians.
02
Conversion Methodology
To convert radians per second to revolutions per minute, follow these steps: 1. Convert radians to revolutions: \( \omega \times \frac{1 \text{ revolution}}{2\pi \text{ radians}} \).2. Convert the time unit from seconds to minutes: Multiply by 60 seconds/minute.
03
Step-by-Step Conversion
Starting with \( 22.0 \, \mathrm{rad/s} \):1. Convert radians to revolutions:\[\frac{22.0 \, \text{rad}}{1 \, \text{s}} \times \frac{1 \, \text{revolution}}{2\pi \, \text{rad}}\]Resulting in:\[\frac{22.0}{2\pi} \, \text{revolutions/s}\]2. Convert seconds to minutes:\[\frac{22.0}{2\pi} \, \text{revolution/s} \times 60 \, \text{s/min}\]Resulting in:\[\frac{22.0 \times 60}{2\pi} \, \text{revolutions/min}\]
04
Final Calculation
Perform the calculation:\[\frac{22.0 \times 60}{2 \pi} \approx 210.0 \, \text{rpm}\]Thus, the angular speed in revolutions per minute is approximately 210.0 \( \text{rpm} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Revolutions per minute (rpm)
Revolutions per minute, abbreviated as rpm, is a unit of angular velocity that indicates the number of complete turns or cycles an object makes in one minute. It is a common unit used to express rotational speed, especially in mechanical and engineering contexts like car engines or rotating discs such as CDs.
By understanding rpm, we can effectively gauge how fast something is spinning, which is crucial in both practical and theoretical scenarios. Since one full circle corresponds to one revolution, clearly knowing how many revolutions occur in a given time frame, like a minute, allows for the calculation of angular distance over time.
By understanding rpm, we can effectively gauge how fast something is spinning, which is crucial in both practical and theoretical scenarios. Since one full circle corresponds to one revolution, clearly knowing how many revolutions occur in a given time frame, like a minute, allows for the calculation of angular distance over time.
Radians per second (rad/s)
Radians per second, often noted as rad/s, is another unit of angular velocity used predominantly in physics to describe how fast an object rotates or revolves relative to its axis. One radian is the angle created when the arc length equals the radius of the circle, and this unit is part of the International System of Units (SI).
The concept of rad/s is significant because it directly ties to basic angular measurements and is useful for calculations involving radii and circular paths. When comparing it to rpm, it might seem less intuitive since we think more naturally in terms like revolutions or rotations, but rad/s provides a more precise measure in terms of the radian-angle that has mathematical richness in calculus and trigonometry.
The concept of rad/s is significant because it directly ties to basic angular measurements and is useful for calculations involving radii and circular paths. When comparing it to rpm, it might seem less intuitive since we think more naturally in terms like revolutions or rotations, but rad/s provides a more precise measure in terms of the radian-angle that has mathematical richness in calculus and trigonometry.
Unit conversion
Converting from one unit of measure to another is an essential skill, especially in scientific fields where precise calculations are necessary. Here, we focus on converting between radians per second and revolutions per minute.
The unit conversion process requires two steps: first, alter the angular measure from radians to revolutions using the fact that one revolution equals 2π radians. This involves dividing the given radians per second by 2π.
The unit conversion process requires two steps: first, alter the angular measure from radians to revolutions using the fact that one revolution equals 2π radians. This involves dividing the given radians per second by 2π.
- Step 1: \( rac{22.0 imes 1}{2\pi} \text{ revolution/s} \)
Circular motion
Circular motion refers to the movement of an object along a circular path or a circular trajectory. In the context of our example, a CD spinning in a player exemplifies circular motion. This type of motion is characterized by continuous rotation around a fixed central point or axis.
Some key concepts associated with circular motion include angular velocity, centripetal force, and centripetal acceleration. Angular velocity directly corresponds to how rapidly the object rotates about the axis, and it's measured in units we've discussed, like rpm or rad/s. Understanding the mechanics of circular motion helps in analyzing systems where rotation plays a vital role, from celestial orbits involving planets to everyday machinery operations.
Some key concepts associated with circular motion include angular velocity, centripetal force, and centripetal acceleration. Angular velocity directly corresponds to how rapidly the object rotates about the axis, and it's measured in units we've discussed, like rpm or rad/s. Understanding the mechanics of circular motion helps in analyzing systems where rotation plays a vital role, from celestial orbits involving planets to everyday machinery operations.
