Problem 9 - College Physics

Question

\(\bullet\) An airplane propeller is rotating at 1900 \(\mathrm{rpm.}\) (a) Compute the propeller's angular velocity in rad/s. (b) How many seconds does it take for the propeller to turn through \(35^{\circ} ?\) (c) If the propeller were turning at \(18 \mathrm{rad} / \mathrm{s},\) at how many rpm would it be turning? (d) What is the period (in seconds) of this propeller?

Step-by-Step Solution

Answer

(a) 198.97 rad/s; (b) 0.0031 s; (c) 171.89 rpm; (d) 0.0316 s.

Step 1: Convert rpm to rad/s

To convert rotations per minute (rpm) to radians per second (rad/s), use the conversion factor: \(1 \text{ rotation} = 2\pi \text{ radians}\). Thus: \[ \frac{1900 \text{ rotations}}{1 \text{ minute}} \times \frac{2\pi \text{ radians}}{1 \text{ rotation}} \times \frac{1 \text{ minute}}{60 \text{ seconds}} = \frac{1900 \times 2\pi}{60} \approx 198.97 \text{ rad/s}. \]

Step 2: Calculate time for 35 degrees

First, convert degrees to radians, knowing that \(1 \text{ degree} = \frac{\pi}{180} \text{ radians}\).\(35^{\circ} = 35 \times \frac{\pi}{180} \approx 0.611 \text{ radians}.\)Using angular velocity \(\omega = 198.97\text{ rad/s},\) find time \(t\) with \(\theta = \omega t\): \[ t = \frac{\theta}{\omega} = \frac{0.611}{198.97} \approx 0.0031 \text{ seconds}. \]

Step 3: Convert 18 rad/s to rpm

To convert rad/s to rpm, reverse the conversion process: \[ 18 \text{ rad/s} \times \frac{1 \text{ rotation}}{2\pi \text{ radians}} \times \frac{60 \text{ seconds}}{1 \text{ minute}} = \frac{18 \times 60}{2\pi} \approx 171.89 \text{ rpm}. \]

Step 4: Determine the period

The period \(T\) is the reciprocal of the angular velocity. It is the time for one complete rotation.Using \(\omega = 198.97 \text{ rad/s}\):\[ T = \frac{2\pi}{\omega} = \frac{2\pi}{198.97} \approx 0.0316 \text{ seconds}. \]

Key Concepts

Understanding Radians Per Second

The concept of angular velocity is critical when investigating rotating objects. Angular velocity expresses how fast an object is rotating. It is often measured in radians per second (rad/s). This unit reflects how many radians an object moves through in one second.
For example, if a propeller spins with an angular velocity of 198.97 rad/s, this means it rotates approximately 198.97 radians every second.
In practical terms, this measurement is part of what allows engineers and technicians to assess how smoothly parts of machines, like airplane propellers, are operating. Rad/s is a fundamental unit in physics and engineering, offering a universal interpretation across platforms using rotational movements. To better understand angular velocity in rad/s, remember:

• 1 full rotation is equal to \( 2\pi \) radians.
• It's a standard measure to express rotational speed.
• Useful in calculating the rotational behavior of various systems.

Degrees to Radians Conversion

Degrees and radians are two units for measuring angles, and converting between them is often necessary in solving rotational motion problems.
Since there are \( 360^{\circ} \) in a full circle and \( 2\pi \) radians correspond to the same full circle, we can set up a conversion. Specifically, one degree is equivalent to \( \frac{\pi}{180} \) radians. This relationship helps in converting an angle in degrees to radians.
For instance, converting \( 35^{\circ} \) to radians would involve:

• \( 35 \times \frac{\pi}{180} = 0.611 \text{ radians} \)

Understanding this conversion is crucial for solving equations needing angles in radians, such as those involving trigonometric functions or angular displacement. It further enables the seamless integration of these units in formulas where radians provide the necessary input.

Exploring Revolutions Per Minute

Revolutions per minute (rpm) is a unit of angular velocity describing how many complete turns occur in one minute.
In many mechanical systems, such as engines and turbines, the operating speed is often given in rpm, as it provides a convenient measure of how fast parts are rotating.
To convert rpm to rad/s or vice versa, use the fact that one revolution equals \( 2\pi \) radians and there are 60 seconds in one minute. Thus, to determine how fast the rotations are in rad/s, apply the formula:

• \( \text{rad/s} = \frac{\text{rpm} \times 2\pi}{60} \)

This is crucial for engineers designing or adjusting mechanical systems, as it allows for compatibility with devices and calculations that require angular measurements in radians rather than revolutions.

Basics of Angular Motion

Angular motion refers to the movement of objects around a circular path or axis. This is characterized by the rotation of an object about a central point or axis within a plane or the third axis in three-dimensional space.
Angular motion is fundamental in physics as it explains how objects behave when they are spinning or revolving, which is extensively applicable in fields like astronomy, robotics, and various engineering disciplines.
Key aspects to understand:

• Angular Displacement: Measured in radians, it is the change in the angle as an object rotates.
• Angular Velocity: Describes how fast an object rotates, usually specified in rad/s or rpm.
• Angular Acceleration: Refers to how the angular velocity changes over time.

By understanding these concepts, one can analyze the behavior of rotating systems, crucial for the design and interpretation of motion in mechanical and dynamic systems.