Problem 28 - Applied Physics

Question

An airplane propeller with blades \(2.00 \mathrm{~m}\) long is rotating at 1150 rpm. (a) Express its angular speed in \(\mathrm{rad} / \mathrm{s}\). (b) Find its angular displacement in \(4.00 \mathrm{~s}\). (c) Find the linear speed (in \(\mathrm{m} / \mathrm{s}\) ) of a point on the end of the blade. (d) Find the linear speed (in \(\mathrm{m} / \mathrm{s}\) ) of a point \(1.00 \mathrm{~m}\) from the end of the blade.

Step-by-Step Solution

Answer

(a) 120.55 rad/s, (b) 482.2 rad, (c) 241.1 m/s, (d) 120.55 m/s.

Step 1: Converting Rotations per Minute to Radians per Second

The angular speed in revolutions per minute (rpm) is given as \(1150\) rpm. To convert this to radians per second (rad/s), use the conversion factor:\[\omega = \frac{1150 \text{ rev/min} \times 2\pi \text{ rad/rev}}{60 \text{ s/min}}\] This simplifies to:\[\omega \approx 120.55 \text{ rad/s}\]

Step 2: Calculating Angular Displacement

To find the angular displacement \(\theta\) after \(4.00\) seconds, use the angular displacement formula:\[\theta = \omega \times t\]where \(\omega = 120.55 \text{ rad/s}\) and \(t = 4.00 \text{ s}\). Substituting in these values gives:\[\theta = 120.55 \times 4.00 = 482.2 \text{ rad}\]

Step 3: Determining Linear Speed at the Blade Tip

The linear speed \(v\) of a point on the tip of the blade (\(r = 2.00 \text{ m}\)) can be found using the formula:\[v = r \times \omega\] Substitute \(r = 2.00 \text{ m}\) and \(\omega = 120.55 \text{ rad/s}\):\[v = 2.00 \times 120.55 \approx 241.1 \text{ m/s}\]

Step 4: Determining Linear Speed 1 Meter from the Blade Tip

To find the linear speed of a point \(1.00 \text{ m}\) from the end, the effective radius will be \(r = 1.00 \text{ m}\). Use the same linear speed formula:\[v = r \times \omega\] Substitute \(r = 1.00 \text{ m}\) and \(\omega = 120.55 \text{ rad/s}\):\[v = 1.00 \times 120.55 \approx 120.55 \text{ m/s}\]

Key Concepts

Angular Speed

Angular speed tells us how fast something rotates or spins. It is expressed in radians per second (rad/s). Imagine spinning your finger around a ball; angular speed measures how quickly you complete circles around it.

To convert rotations per minute (rpm) to radians per second, a simple formula helps:

• 1 revolution equals 2π radians because a full circle is 2π radians.
• There are 60 seconds in a minute, so we divide our result by 60.
• Thus, the formula becomes \(\omega = \frac{1150 \times 2\pi}{60} \\approx 120.55 \, \text{rad/s}\).

Understanding the conversion not only answers the immediate problem but also emphasizes the importance of knowing your units, a key skill in physics.

Angular Displacement

Angular displacement measures the angle through which an object has rotated in a given period. It's similar to linear distance but for rotation, and it's in radians.

To find the angular displacement, use the formula:

• \(\theta = \omega \times t\)
• Where \(\omega = 120.55 \, \text{rad/s}\) and \(t = 4.00 \, \text{s}\).
• The calculation becomes \(\theta = 120.55 \times 4.00 = 482.2 \, \text{rad}\).

This result indicates that over 4 seconds, the object has rotated through 482.2 radians. Visualizing this can be like watching the hands of a clock sweep continuously over time, marking the vast distance covered by the rotation.

Linear Speed

Linear speed is how fast something moves along a path. When something is spinning, like a propeller, points at the edge have linear speed as they travel around the circle.

To calculate the linear speed at the tip of the blade, we use:

• \(v = r \times \omega\)
• For a blade length of \(r = 2.00 \, \text{m}\), substitute \(\omega = 120.55 \, \text{rad/s}\)
• The calculation becomes \(v = 2.00 \times 120.55 \approx 241.1 \, \text{m/s}\).

This means that each point at the tip of the blade moves 241.1 meters in one second. Likewise, 1 meter from the tip, it moves 120.55 meters per second, showing how speed decreases as you move closer to the rotation center.

Conversion of Units

Conversion of units is crucial in rotational dynamics to ensure consistency and accuracy. Converting from radians per second to degrees or minutes necessitates understanding these unit relationships.

Some key conversions include:

• 1 revolution is equivalent to 360 degrees or 2π radians.
• To convert to different time units, consider the number of seconds per minute or hours as required.

We employed these conversions in the exercise to convert 1150 rpm (revolutions per minute) into rad/s, ensuring results are in the desired unit for comparison or further calculations. Mastery of unit conversions helps in adapting to different problem requirements effectively.