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Chapter 8: Problem 61

At offers one approach to solving this problem. The drawing shows the blade of a chain saw. The rotating sprocket tip at the end of the guide bar has a radius of \(4.0 \times 10^{-2} \mathrm{~m}\). The linear speed of a chain link at point \(\mathrm{A}\) is \(5.6 \mathrm{~m} / \mathrm{s}\). Find the angular speed of the sprocket tip in rev/s.

Short Answer

Expert verified
The angular speed is approximately 22.3 rev/s.

Step by step solution

01

Identify Known Values

We are given the linear speed of a chain link at point A, which is \(v = 5.6 \text{ m/s}\), and the radius of the rotating sprocket tip is \(r = 4.0 \times 10^{-2} \text{ m}\). We need to find the angular speed, denoted by \(\omega\).
02

Understand the Relationship Between Linear and Angular Speed

The relationship between linear speed \(v\) and angular speed \(\omega\) is given by the formula: \[ v = r \times \omega \]where \(r\) is the radius of the circular path.
03

Rearrange the Formula to Solve for Angular Speed

To find the angular speed \(\omega\), we can rearrange the formula to:\[ \omega = \frac{v}{r} \]
04

Substitute Known Values into the Formula

Substitute \(v = 5.6 \text{ m/s}\) and \(r = 4.0 \times 10^{-2} \text{ m}\) into the equation:\[ \omega = \frac{5.6}{4.0 \times 10^{-2}} \]
05

Calculate the Angular Speed

Perform the division to find \(\omega\):\[ \omega = \frac{5.6}{0.04} = 140 \text{ rad/s} \]
06

Convert Angular Speed from Rad/s to Rev/s

Since 1 revolution is equal to \(2\pi\) radians, the angular speed in revolutions per second \(\omega_{rev}\) is:\[ \omega_{rev} = \frac{140}{2\pi} = \frac{140}{6.2832} \approx 22.3 \text{ rev/s} \]

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Key Concepts

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

Linear Speed
Linear speed refers to how fast an object is moving along a path. It is a scalar quantity, which simply means it has magnitude but no direction. In this exercise, the linear speed is the rate at which a chain link moves at point A on the chain saw blade. The formula to calculate linear speed is straightforward:
  • \( v = r \times \omega \)
Here, \( v \) is the linear speed, \( r \) is the radius, and \( \omega \) is the angular speed. Linear speed is often measured in meters per second (m/s). Understanding linear speed is crucial because it helps us relate how fast a point on a rotating object is moving in a straight line.
Sprocket Tip
The sprocket tip is the rotating component found at the end of the chain saw鈥檚 guide bar. It serves as a vital part of the mechanism that helps guide the chain around the bar. When we discuss the sprocket tip, we focus especially on its radius. This radius is vital for several calculations, such as determining the linear speed of the chain link. In the given problem, this radius is specified to be \(4.0 \times 10^{-2} \) meters.Having an accurate measurement of the sprocket tip's radius is crucial, as it directly impacts the angular speed calculation. As the radius influences the entire motion dynamics, understanding this can help one efficiently translate between linear and angular speeds.
Circular Motion
Circular motion describes the movement of an object along a circular path. This type of motion has both a distance component and a rotational component. A sprocket tip moving in a circular path is a perfect example of circular motion. The blade's chain uses this principle to make cutting possible. Circular motion requires the presence of two factors:
  • A fixed axis, around which the motion occurs.
  • A constant radius, the distance from the axis to the moving object.
Understanding circular motion is essential for determining the relationships between linear speed and angular speed, as well as recognizing how forces act in rotational dynamics.
Radians to Revolutions Conversion
Radians and revolutions are both units used to measure angles in rotation. One full revolution equals the angle at which a point completes a full circle around a center. To convert from radians to revolutions:
  • Recognize that one revolution equals \(2\pi\) radians.
  • Use the conversion formula: \[ \omega_{\text{rev}} = \frac{\omega_{\text{rad/s}}}{2\pi} \]
In our exercise, the angular speed in radians per second has to be converted to revolutions per second. The process is crucial for understanding how quickly the sprocket tip spins, in terms humans find intuitive and natural: turns per second.

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Most popular questions from this chapter

A thin rod (length \(=1.50 \mathrm{~m}\) ) is oriented vertically, with its bottom end attached to the floor by means of a frictionless hinge. The mass of the rod may be ignored, compared to the mass of an object fixed to the top of the rod. The rod, starting from rest, tips over and rotates downward. (a) What is the angular speed of the rod just before it strikes the floor? (Hint: Consider using the principle of conservation of mechanical energy.) (b) What is the magnitude of the angular acceleration of the rod just before it strikes the floor?

A CD has a playing time of 74 minutes. When the music starts, the \(\mathrm{CD}\) is rotating at an angular speed of 480 revolutions per minute (rpm). At the end of the music, the \(\mathrm{CD}\) is rotating at \(210 \mathrm{rpm}\). Find the magnitude of the average angular acceleration of the \(\mathrm{CD}\). Express your answer in \(\mathrm{rad} / \mathrm{s}^{2}\)

A baseball pitcher throws a baseball horizontally at a linear speed of \(42.5 \mathrm{~m} / \mathrm{s}\) (about 95 \(\mathrm{mi} / \mathrm{h}\) ). Before being caught, the baseball travels a horizontal distance of \(16.5 \mathrm{~m}\) and rotates through an angle of 49.0 rad. The baseball has a radius of \(3.67 \mathrm{~cm}\) and is rotating about an axis as it travels, much like the earth does. What is the tangential speed of a point on the "equator" of the baseball?

Our sun rotates in a circular orbit about the center of the Milky Way galaxy. The radius of the orbit is \(2.2 \times 10^{20} \mathrm{~m},\) and the angular speed of the sun is \(1.2 \times 10^{-15} \mathrm{rad} / \mathrm{s}\). (a) What is the tangential speed of the sun? (b) How long (in years) does it take for the sun to make one revolution around the center?

A propeller is rotating about an axis perpendicular to its center, as the drawing shows. The axis is parallel to the ground. An arrow is fired at the propeller, travels parallel to the axis, and passes through one of the open spaces between the propeller blades. The vertical drop of the arrow may be ignored. There is a maximum value for the angular speed \(\omega\) of the propeller beyond which the arrow cannot pass through an open space without being struck by one of the blades. (a) If the arrow is to pass through an open space, does it matter if the arrow is aimed closer to or farther away from the axis (see points \(\mathrm{A}\) and \(\mathrm{B}\) in the drawing, for example)? Explain. (b) Does the maximum value of \(\omega\) increase, decrease, or remain the same with increasing arrow speed \(v ?\) Why? (c) Does the maximum value of \(\omega\) increase, decrease, or remain the same with increasing arrow length \(L\) ? Justify your answer.
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