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Chapter 10: Problem 32

The outer edge of a rotating Frisbee with a diameter of \(29 \mathrm{cm}\) has a linear speed of \(3.7 \mathrm{m} / \mathrm{s}\). What is the angular speed of the Frisbee?

Short Answer

Expert verified
The angular speed of the Frisbee is approximately 25.52 rad/s.

Step by step solution

01

Understand the Relationship

The relationship between linear speed (v), angular speed (蠅), and radius (r) of a rotating object is given by the formula: \( v = \omega \times r \). Here, \( v \) is the linear speed, and \( r \) is the radius of the Frisbee. We need to find \( \omega \).
02

Calculate the Radius

Since the diameter of the Frisbee is \( 29 \text{ cm} \), the radius \( r \) is half of the diameter. Convert this radius to meters, as our linear speed is given in meters per second. \( r = \frac{29}{2} \text{ cm} = 14.5 \text{ cm} = 0.145 \text{ m} \).
03

Rearrange the Formula

Rearrange the formula to solve for angular speed \( \omega \): \( \omega = \frac{v}{r} \).
04

Insert Values into Formula

Substitute the known values into the rearranged formula: \( \omega = \frac{3.7 \text{ m/s}}{0.145 \text{ m}} \).
05

Perform the Calculation

Calculate \( \omega \) using the values provided: \( \omega = \frac{3.7}{0.145} \approx 25.52 \text{ rad/s} \).
06

Conclude with Angular Speed

The angular speed of the Frisbee is approximately \( 25.52 \text{ rad/s} \).

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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 moves along a path. Imagine how quickly the edge of the Frisbee travels as it rotates in the air. This is measured in meters per second (m/s) to express how far the edge moves every second.
Understanding linear speed is important because it directly relates to angular speed through the radius of rotational motion. If we know the linear speed and radius, we can calculate how fast something rotates (angular speed).
  • Linear speed ( \( v \)) is linked to how much distance a point on the outer edge of a rotating object covers over a certain time.
  • In our Frisbee problem, the linear speed is given as 3.7 m/s.
  • It helps us find the angular speed, which describes how fast the object is rotating around its center.
Understanding this concept helps us appreciate how parts of rotating bodies move and how fast they can cover distances.
Radius
The radius is the distance from the center of a circle to any point on its outer edge. In the case of our rotating Frisbee, this is half of its diameter.
To transition your measurements smoothly from centimeters to meters (the standard unit in physics), simply divide the diameter by two to find the radius in centimeters, then convert to meters if needed.
  • The given diameter is 29 cm, so the radius is 14.5 cm.
  • In meters, this is 0.145 m, which is crucial for using in the formula relating linear and angular speeds.
This radius helps us determine how far points on the edge are from the center, impacting how fast they move in linear speed.
Rotational Motion
Rotational motion is when an object spins around an axis. Imagine how the Frisbee spins as it's thrown! This type of motion links linear speed and angular speed.
The key relationship is through the formula linking them: \( v = \omega \times r \), where \( \omega \) (angular speed) explains how quickly the Frisbee rotates.
  • Angular speed (\( \omega \)) is given in radians per second (rad/s), showing how many angles (radians) it sweeps out per second.
  • In our example, by rearranging the formula to \( \omega = \frac{v}{r} \), we found the Frisbee's angular speed to be roughly 25.52 rad/s.
  • This computation highlights the connection between how fast the Frisbee spins and how fast points along its edge move.
This interplay of rotational and linear aspects enriches our understanding of how objects move in a circular path, especially useful in analyzing everyday rotating objects like wheels or gears.

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Most popular questions from this chapter

A 1.3 -kg block is tied to a string that is wrapped a round the rim of a pulley of radius \(7.2 \mathrm{cm}\). The block is released from rest. (a) Assuming the pulley is a uniform disk with a mass of \(0.31 \mathrm{kg},\) find the speed of the block after it has fallen through a height of \(0.50 \mathrm{m}\). (b) If a small lead weight is attached near the rim of the pulley and this experiment is repeated, will the speed of the block increase, decrease, or stay the same? Explain.

The world's tallest building is the Taipei 101 Tower in Taiwan, which rises to a height of \(508 \mathrm{m}(1667 \mathrm{ft})\). (a) When standing on the top floor of the building, is your angular speed due to the Earth's rotation greater than, less than, or equal to your angular speed when you stand on the ground floor? (b) Choose the hest explanation from among the following: I. The angular speed is the same at all distances from the axis of rotation. II. At the top of the building you are farther from the axis of rotation and hence you have a greater angular speed. III. You are spinning faster when you are closer to the axis of rotation.

Express the angular velocity of the second hand on a clock in the following units: (a) rev / hr, (b) \(\operatorname{deg} / \mathrm{min},\) and (c) \(\mathrm{rad} / \mathrm{s}\).

A compact disk (CD) speeds up uniformly from rest to 310 rpm in \(3.3 \mathrm{s}\). (a) Describe a strategy that allows you to calculate the number of revolutions the CD makes in this time. (b) Use your strategy to find the number of revolutions.

A spot of paint on a bicycle tire moves in a circular path of radius \(0.33 \mathrm{m}\). When the spot has traveled a linear distance of \(1.95 \mathrm{m},\) through what angle has the tire rotated? Give your answer in radians.
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