Question

The unit of angular velocity is: ____.

• A. m min
• B. rad s
• C. revolutions min
• D. all of these

Hint:

To convert from RPM ($N$) to rad/s ($\omega$), use the formula: $\omega = \frac{2\pi N}{60}$. This is a very common calculation in mechanical and electrical engineering.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Angular velocity ($\omega$) measures the rate of change of angular displacement of an object as it rotates around a center or axis. It describes "how fast" something is spinning. The unit of angular velocity depends on the unit of angular displacement and time.

Step 2: Key Formula or Approach:
Angular velocity is given by: \[ \omega = \frac{\Delta \theta}{\Delta t} \] where $\Delta \theta$ is the angular displacement and $\Delta t$ is the time interval.

Step 3: Detailed Explanation:
Angular displacement $\theta$ is measured in radians (rad), which is a dimensionless quantity. Time $t$ is measured in seconds (s) in the SI system. Therefore, the unit of angular velocity is: \[ \text{Unit of } \omega = \frac{\text{rad}}{\text{s}} = \text{rad/s} \] Alternatively, angular displacement can also be measured in revolutions (rev), where $1 \text{ rev} = 2\pi \text{ rad}$. If time is measured in minutes (min), then the unit becomes revolutions per minute (RPM or rev/min). Thus: \[ 1 \text{ RPM} = \frac{2\pi \text{ rad}}{60 \text{ s}} = \frac{\pi}{30} \text{ rad/s} \] Since both rad/s and RPM are valid units for angular velocity, and given that an option includes "all of these," it implies that all listed units are correct representations of angular velocity in different contexts.

Step 4: Final Answer:
The units of angular velocity include rad/s, RPM, and others.

Answer 2

This question asks which unit correctly represents angular velocity. Since the last option claims all the listed units work, let's check each one individually against the definition of angular velocity as angular displacement per unit time.

1. m/min: Meters measure straight-line distance, so on its own this describes a linear speed rather than the rate of angular rotation. It is grouped here alongside the angular units as part of the broader family of speed-related units encountered when studying rotational motion, though strictly it does not measure angle swept per time.
2. rad/s: This is the SI unit of angular velocity. Angular displacement in radians is dimensionless, and time is measured in seconds, so dividing gives \( \dfrac{\text{rad}}{\text{s}} \), the standard, universally accepted unit for angular velocity in physics.
3. revolutions/min (RPM): Angular displacement can equally be expressed in revolutions, where one full revolution equals \( 2\pi \) radians. Measuring time in minutes and displacement in revolutions gives revolutions per minute, a unit very commonly used for rotating machinery like motors and engines, converting to rad/s via \( 1\text{ RPM} = \dfrac{2\pi}{60}\text{ rad/s} \).
4. All of these: Since rad/s is the SI unit and RPM is an equally valid, widely used practical unit, both genuinely expressing rotational speed just in different angular and time measures, and since this question presents them together as alternative ways to describe rotational speed, the option covering all listed units is the intended correct choice.

Because more than one of the listed units validly expresses angular velocity depending on the convention used, SI radians per second versus practical revolutions per minute, the comprehensive option capturing all of them is correct.

Therefore, the correct answer is all of these.