Problem 18 - Applied Physics

Question

A rotor completes \(50.0\) revolutions in \(3.25 \mathrm{~s}\). Find its angular speed (a) in rev/s. (b) in rpm. (c) in rad/s.

Step-by-Step Solution

Answer

Angular speeds are 15.38 rev/s, 922.8 rpm, and 96.64 rad/s.

Step 1: Convert Revolutions to Revolutions per Second (rev/s)

To find the angular speed in revolutions per second (rev/s), divide the total number of revolutions by the time in seconds.Given:- Total revolutions: 50.0 rev- Time: 3.25 secondsFormula: \[ \text{Angular speed in rev/s} = \frac{\text{Total revolutions}}{\text{Time in seconds}} \]Calculation:\[ \text{Angular speed in rev/s} = \frac{50.0 \text{ rev}}{3.25 \text{ s}} = 15.38 \text{ rev/s} \]

Step 2: Convert Revolutions per Second to Revolutions per Minute (rpm)

To convert the angular speed from revolutions per second to revolutions per minute, multiply the result from Step 1 by 60 (there are 60 seconds in a minute).Formula:\[ \text{Angular speed in rpm} = \text{Angular speed in rev/s} \times 60 \]Calculation:\[ \text{Angular speed in rpm} = 15.38 \text{ rev/s} \times 60 = 922.8 \text{ rpm} \]

Step 3: Convert Revolutions per Second to Radians per Second (rad/s)

To convert the angular speed from revolutions per second to radians per second, multiply the result from Step 1 by \(2\pi\) since one revolution equals \(2\pi\) radians.Formula:\[ \text{Angular speed in rad/s} = \text{Angular speed in rev/s} \times 2\pi \]Calculation:\[ \text{Angular speed in rad/s} = 15.38 \text{ rev/s} \times 2\pi = 96.64 \text{ rad/s} \]

Key Concepts

Revolutions per Second

Understanding how fast something is spinning can be described using revolutions per second (rev/s). This is a measure of how many complete spins or turns an object makes in one second. To calculate it, you simply divide the total number of revolutions by the total time in seconds.
For example, if a rotor completes 50 revolutions in 3.25 seconds, the angular speed would be:

• Using the formula: \( \text{rev/s} = \frac{\text{Total revolutions}}{\text{Time in seconds}} \)
• This gives \( \frac{50}{3.25} = 15.38 \text{ rev/s} \)

This straightforward calculation helps in determining the speed at which an object is rotating.

Revolutions per Minute

Revolutions per minute (rpm) offers another common way to express angular speed, especially in situations such as engine speeds or other machines. To convert from revolutions per second to revolutions per minute, you multiply by 60, because there are 60 seconds in one minute.
For instance:

• If your object's speed is 15.38 rev/s, you calculate rpm by multiplying: \( 15.38 \times 60 = 922.8 \text{ rpm} \)

Using rpm can provide a more intuitive understanding of speed when minutes are a more relevant unit of time for the context.

Radians per Second

Radians per second (rad/s) is another way to measure angular speed. This unit is particularly useful in physics and engineering because it aligns with the standard unit of rotational measure, the radian. One full revolution equates to \(2\pi\) radians.
To convert from revolutions per second to radians per second:

• Multiply the rev/s value by \(2\pi\).
• For our example: \(15.38 \text{ rev/s} \times 2\pi = 96.64 \text{ rad/s} \).

Radians offer a way to incorporate rotational motion in calculations more directly related to circular arc length and other geometric considerations.

Unit Conversion

Unit conversion is a vital skill when dealing with different measures of speed such as rev/s, rpm, and rad/s. Knowing how to switch between units ensures that calculations are accurate regardless of which measure is used.
Some common conversions include:

• From rev/s to rpm by multiplying by 60.
• From rev/s to rad/s by multiplying by \(2\pi\).

These conversions help translate rotational speed into forms that are suited for varying contexts and applications.

Rotational Motion

Rotational motion describes the motion of a body around a center or axis. It is an essential concept in physics and appears frequently in engineering, from the operation of motors to the orbits of celestial bodies.
Key attributes of rotational motion include:

• Angular speed, expressed in units like rev/s, rpm, and rad/s.
• Angular displacement, which is the angle through which a point or line has been rotated in a specified sense about a specified axis.

Understanding these principles allows us to analyze and predict the behavior of spinning objects, from everyday machinery to astronomical phenomena.