Question

If an airplane propeller of radius \(5.0 \mathrm{ft}(=1.5 \mathrm{~m})\) rotates at 2000 rev/min and the airplane is propelled at a ground speed of \(300 \mathrm{mi} / \mathrm{h}(=480 \mathrm{~km} / \mathrm{h})\), what is the speed of a point on the tip of the propeller, as seen by \((a)\) the pilot and \((b)\) an observer on the ground? Assume that the plane's velocity is parallel to the propeller's axis of rotation.

Short answer

The speed of a point on the tip of the propeller, as seen by (a) the pilot is 314.16 m/s and (b) an observer on the ground is 448.27 m/s.

Step-by-step solution

Convert the propeller rotations to velocity

First, we need to find out the speed of the tip of the propeller from the pilot's perspective. This is equal to the tangential speed, which is controlled by the rotation speed. The tangential speed \(v_t\) is calculated by the formula \(v_t = r * \omega\), where \(r\) is the radius and \(\omega\) is the angular speed. Let's convert 2000 revolutions per minute to radians per second: \( \omega = 2000 \, rev/min * (2\pi \, rad/rev) * (1 \, min/60 \, s) = 209.44 \, rad/s\). Therefore, the speed of the tip of the propeller from the pilot's perspective is \( v_t = 1.5 \, m * 209.44 \, rad/s = 314.16 \, m/s\).

Convert the plane's velocity

The plane's velocity needs to be converted from miles per hour to meters per second, since we're asked to provide the solution in the same units as the speed of the propeller tip. Therefore, \(v = 300 \, mi/h * (1609.34 \, m/mi) * (1 \, h/3600 \, s) = 134.11 \, m/s\).

Calculate the speed of the propeller tip as seen by an observer on the ground

Finally, to find the speed of the propeller tip from the perspective of an observer on the ground, we need to realize that the observer on the ground also sees the airplane's velocity. So, the total speed of the tip of the propeller, as seen by an observer on the ground, is the sum of the plane's speed and the tip's speed relative to the pilot: \(v_t' = v + v_t = 134.11 \, m/s + 314.16 \, m/s = 448.27 \, m/s\).

Key concepts

Angular Velocity

The concept of angular velocity is vital in understanding rotational motion.
It refers to how quickly an object rotates or revolves around a central point and is often measured in radians per second.- In our exercise, the propeller's angular velocity is given in revolutions per minute (rev/min).
To convert this to a more conventional unit like radians per second (rad/s), we use the relationship that one revolution equals \(2\pi\) radians.- Thus, for a propeller spinning at 2000 rev/min, we multiply by \(2\pi\) and divide by 60 because there are 60 seconds in a minute.This conversion helps in calculating the tangential speed, which is essential for comparing speeds relative to both the pilot and the ground observer.

Velocity Conversion

Being able to convert between different velocity units is a necessary skill, especially when dealing with problems involving motion.
Velocity in physics can often be presented in units such as miles per hour (mi/h), kilometers per hour (km/h), or meters per second (m/s). - For our scenario, the airplane's speed is initially given in miles per hour.
Since the angular speed of the propeller was converted to meters per second, we need all velocity measurements in the same unit to make accurate calculations. - To convert miles per hour to meters per second, we use the conversion factors: 1 mile equals 1609.34 meters and 1 hour equals 3600 seconds.
This ensures that all velocity terms can be directly compared or combined as needed in the problem.

Propeller Motion

Propeller motion involves understanding how a rotating object interacts with other motion, such as the movement of the airplane itself.
- The speed of a point on the edge of the propeller, called the tip speed, can be split into various perspectives: - **From the Pilot's View**: The pilot sees only the rotational motion, not incorporating the forward movement of the plane.
This is the tangential speed, calculated with the radius and the angular velocity. - **From the Observer's View**: This incorporates both the rotational speed and the linear speed of the plane.
The observer on the ground perceives the sum of the airplane's velocity plus the tangential speed of the propeller tip. Understanding these perspectives is crucial in real-world applications like aviation, where observers might interpret speeds differently based on their reference frame.