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Chapter 4: Problem 58

A rotating fan completes 1200 revolutions every minute. Consider the tip of a blade, at a radius of \(0.15 \mathrm{~m}\). (a) Through what distance does the tip move in one revolution? What are (b) the tip's speed and (c) the magnitude of its acceleration? (d) What is the period of the motion?

Short Answer

Expert verified
(a) \(0.30\pi \ m\), (b) \(6\pi \ m/s\), (c) \(240\pi^2 \ m/s^2\), (d) \(0.05 \ s\)

Step by step solution

01

Determine the Distance per Revolution

The distance the tip moves in one revolution is the circumference of the circle formed by the blade. The formula for the circumference is \( C = 2\pi r \). Here \( r = 0.15 \ m \), so:\[C = 2 \pi \times 0.15 \ m = 0.30\pi \ m\]Thus, the distance the tip moves in one revolution is \(0.30\pi \ m\).
02

Calculate the Tip's Speed

The speed of the tip is the distance covered in one minute divided by the time. Since the fan completes 1200 revolutions per minute, the distance covered is:\[1200 \times 0.30 \pi \ m = 360\pi \ m\]The speed \( v \) is thus:\[v = \frac{360\pi \ m}{60 \ s} = 6\pi \ m/s\]
03

Calculate the Magnitude of Acceleration

The acceleration of the tip is centripetal, given by the formula \( a = \frac{v^2}{r} \). Using \( v = 6\pi \ m/s \) and \( r = 0.15 \ m \):\[a = \frac{(6\pi)^2}{0.15} = \frac{36\pi^2}{0.15} \ m/s^2\]Simplifying further:\[a = 240\pi^2 \ m/s^2\]
04

Determine the Period of the Motion

The period \( T \) is the time taken for one complete revolution. Since there are 1200 revolutions in one minute (60 seconds), the period is:\[T = \frac{60 \ s}{1200} = 0.05 \ s\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Centripetal Acceleration
Centripetal acceleration is a key concept in rotational motion. It is the acceleration that keeps an object moving in a circular path. Instead of pushing the object outward, it keeps pulling the object towards the center of the circle. This means the direction of centripetal acceleration is always towards the center of the circle.

To calculate centripetal acceleration (\[a\]), we use the formula:
  • \[a = \frac{v^2}{r}\],where:
    • \(v\) is the speed of the object, and
    • \(r\) is the radius of the circular path.
In the context of a fan blade tip, centripetal acceleration ensures the tip continues moving in a circle rather than flying off in a straight line. The magnitude of this acceleration can significantly impact the stability and performance of rotating systems like fans.
Circumference Calculation
Circumference is an essential measurement for circular paths. It gives us the total distance around a circle. For any rotating object, knowing the circumference helps us understand how far the object travels in one complete circle.

The formula for circumference (\[C\]) is:
  • \[C = 2\pi r\],where:
    • \(r\) is the radius of the circle.
In our exercise, the fan blade forms a circle with a radius of 0.15 meters, resulting in a circumference of \(0.30\pi\) m. Understanding this helps in calculating the distance travelled by the tip of the blade in one full rotation.
Revolution Period
The revolution period (\[T\]) is the time taken to complete one full cycle around a circular path. In rotational motion, the period helps us understand how quickly an object is revolving.

The formula to determine the period is:
  • \[T = \frac{1}{f}\],where:
    • \(f\) is the frequency, the number of cycles per unit time.
For example, if a fan completes 1200 revolutions in one minute, the period is \(0.05\) seconds. This short period tells us the wheel is spinning very fast, making frequent complete cycles in a short amount of time.
Speed of Rotating Objects
The speed of a rotating object describes how fast the object moves along its circular path. It links the distance an object travels to the time taken. In rotational dynamics, it鈥檚 the speed at the circumference of the object's rotational path.

The formula for speed (\[v\]) is:
  • \[v = \frac{d}{t}\],where:
    • \(d\) is the distance traveled, and
    • \(t\) is the time taken.
For a fan completing 1200 revolutions per minute, the blade tip travels a distance of \(360\pi\) m per minute, yielding a speed of \(6\pi\) meters per second. This rapid speed indicates a fast rotation, typical of electric fans.

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Most popular questions from this chapter

A particle leaves the origin with an initial velocity \(\vec{v}=(3.00 \hat{\mathrm{i}}) \mathrm{m} / \mathrm{s}\) and a constant acceleration \(\vec{a}=(-1.00 \hat{\mathrm{i}}-\) \(0.500 \hat{\mathrm{j}}) \mathrm{m} / \mathrm{s}^{2}\). When it reaches its maximum \(x\) coordinate, what are its (a) velocity and (b) position vector?

The position \(\vec{r}\) of a particle moving in an \(x y\) plane is given by \(\vec{r}=\left(2.00 t^{3}-5.00 t\right) \hat{\mathrm{i}}+\left(6.00-7.00 t^{4}\right) \hat{\mathrm{j}}\), with \(\vec{r}\) in meters and \(\mathrm{t}\) in seconds. In unit-vector notation, calculate (a) \(\vec{r},(\mathrm{~b}) \vec{v}\), and \((\mathrm{c}) \vec{a}\) for \(t=2.00 \mathrm{~s}\) (d) What is the angle between the positive direction of the \(x\) axis and a line tangent to the particle's path at \(t=2.00 \mathrm{~s}\) ?

A light plane attains an airspeed of \(500 \mathrm{~km} / \mathrm{h}\). The pilot sets out for a destination \(800 \mathrm{~km}\) due north but discovers that the plane must be headed \(20.0^{\circ}\) east of due north to fly there directly. The plane arrives in \(2.00 \mathrm{~h}\). What were the (a) magnitude and (b) direction of the wind velocity?

A ball rolls horizontally off the top of a stairway with a speed of \(1.52 \mathrm{~m} / \mathrm{s}\). The steps are \(20.3 \mathrm{~cm}\) high and \(20.3 \mathrm{~cm}\) wide. Which step does the ball hit first?

A baseball leaves a pitcher's hand horizontally at a speed of \(161 \mathrm{~km} / \mathrm{h}\). The distance to the batter is \(18.3 \mathrm{~m}\). (a) How long does the ball take to travel the first half of that distance? (b) The second half? (c) How far does the ball fall freely during the first half? (d) During the second half? (e) Why aren't the quantities in (c) and (d) equal?
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