Chapter 9: Problem 2
An airplane propeller is rotating at 1900 rpm (rev/min). (a) Compute the propeller's angular velocity in rad/s. (b) How many seconds does it take for the propeller to turn through 35 \(^\circ\)?
Short Answer
Expert verified
(a) \(\frac{190\pi}{3}\) rad/s; (b) approximately 0.0031 seconds.
Step by step solution
01
Understand the Problem
We need to find two things: (a) the angular velocity of a propeller given in rotations per minute (rpm), and (b) the time it takes for the propeller to rotate through a certain angle in degrees. Angular velocity is typically expressed in radians per second, and we need to convert degrees to radians for part (b).
02
Convert Rotations to Radians for Angular Velocity
One full rotation is equivalent to a complete circle, or \(360^\circ\), which is \(2\pi\) radians. Given the propeller is rotating at 1900 rpm, we first convert this to radians per minute: \(1900 \text{ rotations/min} \times 2\pi \text{ radians/rotation} = 3800\pi \text{ radians/min}\).
03
Convert Minutes to Seconds for Angular Velocity
Next, convert the angular velocity from radians per minute to radians per second. Since there are 60 seconds in a minute: \(\frac{3800\pi \text{ radians/min}}{60 \text{ seconds/min}} = \frac{380\pi}{6} \text{ radians/sec}\). Simplifying, we get: \(\frac{190\pi}{3} \text{ radians/sec}\).
04
Convert 35 Degrees to Radians
To calculate how long it takes for the propeller to rotate through \(35^\circ\), first convert degrees to radians. Since \(180^\circ = \pi\) radians, we use the conversion: \(35^\circ \times \frac{\pi}{180^\circ} = \frac{35\pi}{180} = \frac{7\pi}{36} \text{ radians}\).
05
Calculate Time to Turn Through a Given Angle
Using the angular velocity found in Step 3, calculate the time to rotate through \(\frac{7\pi}{36}\) radians with an angular velocity of \(\frac{190\pi}{3}\text{ rad/s}\). Using the formula: \(\theta = \omega t\), we rearrange to find \(t = \frac{\theta}{\omega}\). Plugging in values: \(t = \frac{\frac{7\pi}{36}}{\frac{190\pi}{3}}\). This simplifies to \(t = \frac{7 \times 3}{190 \times 36} = \frac{21}{6840} = \frac{1}{325.71}\) seconds.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Radians
Radians are a way of measuring angles based on the radius of a circle. When you think of a circle, it has 360 degrees. But instead of breaking it down into degrees, we think about how many times the radius fits along the circumference. This is where radians come in.
- One full circle is always equal to \(2\pi\) radians.
- So, \(360^\circ\) equals \(2\pi\) radians.
- Half a circle, or \(180^\circ\), is \(\pi\) radians.
Degrees to Radians Conversion
Converting degrees to radians is quite straightforward, and it's an essential skill in trigonometry and physics. This conversion is based on the fact that \(180^\circ\) equals \(\pi\) radians. So, to convert any angle from degrees to radians, you multiply it by \(\frac{\pi}{180}\).
Here's a simple method:
Here's a simple method:
- Take the degree measure, let's say it's \(x^\circ\).
- Multiply by \(\frac{\pi}{180}\) to get the radians.
- For example, \(35^\circ\times \frac{\pi}{180} = \frac{7\pi}{36}\) radians.
Time Calculation in Rotational Motion
When dealing with rotational motion, calculating time involves understanding the relationship between the angle of rotation and the angular velocity. The core formula here is: \[\theta = \omega t\]
Where:
Where:
- \(\theta\) is the angle in radians,
- \(\omega\) is the angular velocity in radians per second,
- \(t\) is time in seconds.
Propeller Rotation Mechanics
The rotation of an airplane propeller is a practical example of rotational motion. Understanding how quickly it spins is essential for both engineers and pilots. Propeller rotation speed is often given in revolutions per minute (rpm), which needs to be converted into standard units for practical use. Here's how:
- Convert revolutions per minute to radians per second to find angular velocity.
- Understand that 1 rotation equals \(2\pi\) radians.
- Use the conversion factor between time - minutes to seconds.