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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.
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.

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Most popular questions from this chapter

A uniform, solid disk with mass \(m\) and radius \(R\) is pivoted about a horizontal axis through its center. A small object of the same mass \(m\) is glued to the rim of the disk. If the disk is released from rest with the small object at the end of a horizontal radius, find the angular speed when the small object is directly below the axis.

As an intern at an engineering firm, you are asked to measure the moment of inertia of a large wheel for rotation about an axis perpendicular to the wheel at its center. You measure the diameter of the wheel to be 0.640 m. Then you mount the wheel on frictionless bearings on a horizontal frictionless axle at the center of the wheel. You wrap a light rope around the wheel and hang an 8.20-kg block of wood from the free end of the rope, as in Fig. E9.45. You release the system from rest and find that the block descends 12.0 m in 4.00 s. What is the moment of inertia of the wheel for this axis?

An electric fan is turned off, and its angular velocity decreases uniformly from 500 rev/min to 200 rev/min in 4.00 s. (a) Find the angular acceleration in rev/s\(^2\) and the number of revolutions made by the motor in the 4.00-s interval. (b) How many more seconds are required for the fan to come to rest if the angular acceleration remains constant at the value calculated in part (a)?

While riding a multispeed bicycle, the rider can select the radius of the rear sprocket that is fixed to the rear axle. The front sprocket of a bicycle has radius 12.0 cm. If the angular speed of the front sprocket is 0.600 rev/s, what is the radius of the rear sprocket for which the tangential speed of a point on the rim of the rear wheel will be 5.00 m/s? The rear wheel has radius 0.330 m.

A turntable rotates with a constant 2.25 rad/s\(^2\) angular acceleration. After 4.00 s it has rotated through an angle of 30.0 rad. What was the angular velocity of the wheel at the beginning of the 4.00-s interval?
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