Problem 2 - University Physics with Modern Physics

Question

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 \(^\circ\)?

Step-by-Step Solution

Answer

(a) \(\frac{190\pi}{3}\) rad/s; (b) approximately 0.0031 seconds.

Step 1: 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).

Step 2: 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: \(1900 \text{ rotations/min} \times 2\pi \text{ radians/rotation} = 3800\pi \text{ radians/min}\).

Step 3: 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: \(\frac{190\pi}{3} \text{ radians/sec}\).

Step 4: 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: \(35^\circ \times \frac{\pi}{180^\circ} = \frac{35\pi}{180} = \frac{7\pi}{36} \text{ radians}\).

Step 5: 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.

Key Concepts

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.

Radians provide a direct relationship between the angle and the radius, offering simplicity in calculations, particularly for circular motion.

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:

• 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.

This conversion allows you to work easily with angles in various mathematical and physical applications, especially when calculating rotation speeds.

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:

• \(\theta\) is the angle in radians,
• \(\omega\) is the angular velocity in radians per second,
• \(t\) is time in seconds.

To find time \(t\), rearrange the formula to:\[ t = \frac{\theta}{\omega}\]This means that if you know the angle and the angular velocity, you can easily find how long it takes for an object like a propeller to complete a certain rotation.

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.

This conversion and understanding allow for better control and analysis of propeller performance in diverse flight conditions.