/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 16 A basketball player is balancing... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 16

A basketball player is balancing a spinning basketball on the tip of his finger. The angular velocity of the ball slows down from 18.5 to \(14.1 \mathrm{rad} / \mathrm{s} .\) During the slow-down, the angular displacement is 85.1 rad. Determine the time it takes for the ball to slow down.

Short Answer

Expert verified
The time taken for the ball to slow down is approximately 5.22 seconds.

Step by step solution

01

Understand What is Given

The initial angular velocity \( \omega_i \) is \( 18.5 \, \text{rad/s} \), the final angular velocity \( \omega_f \) is \( 14.1 \, \text{rad/s} \), and the angular displacement \( \theta \) is \( 85.1 \, \text{rad} \). We need to find the time \( t \) it takes for these changes to occur.
02

Use the Kinematic Equation for Angular Motion

In angular motion, we can use the equation: \[ \theta = \frac{1}{2} (\omega_i + \omega_f) \cdot t \] which relates angular displacement \( \theta \), initial angular velocity \( \omega_i \), final angular velocity \( \omega_f \), and time \( t \). We will solve for \( t \).
03

Solve the Equation for Time

Rearrange the equation to isolate \( t \): \[ t = \frac{2\theta}{\omega_i + \omega_f} \]Substitute the known values: \( \theta = 85.1 \, \text{rad} \), \( \omega_i = 18.5 \, \text{rad/s} \), and \( \omega_f = 14.1 \, \text{rad/s} \).
04

Calculate the Time

Plug in the values: \[ t = \frac{2 \times 85.1}{18.5 + 14.1} \]Calculate the denominator: \( 18.5 + 14.1 = 32.6 \, \text{rad/s} \).Calculate the numerator: \( 2 \times 85.1 = 170.2 \).Now calculate \( t \): \[ t = \frac{170.2}{32.6} \approx 5.22 \, \text{s} \].

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Key Concepts

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

Angular Velocity
Angular velocity is a key concept in rotational motion. It reflects how fast an object rotates around a specific point, like how quickly a basketball spins on a player's finger. The measure of angular velocity is typically in radians per second (rad/s). Angular velocity can change, especially if a constant force is applied or the conditions change, which is exactly what happens when the basketball slows down. In the original exercise, the angular velocity of the basketball slows from 18.5 rad/s to 14.1 rad/s. That slowing can be caused by various factors, such as air resistance or lack of force from the finger.

The equation for constant angular velocity looks similar to linear velocity equations. However, it incorporates an additional parameter called the angular displacement to determine the rate of rotation over a period.

For dynamic scenarios where angular velocity changes like in the exercise, we use kinematic equations to understand the object's motion over time.
Angular Displacement
Angular displacement measures the amount through which a point or a line has been rotated in a specified sense about a specified axis. Often measured in radians, angular displacement gives us a clearer picture of the rotational movement by quantifying how many rotations or fraction of rotations an object completes.

In the context of the basketball spinning, the angular displacement is given as 85.1 rad. This means as the basketball slows down, it covers this angular distance. Angular displacement is distinct from linear displacement, as it deals with rotation rather than straight-line motion.

Tracking angular displacement is crucial in kinematics for calculating variables such as time, final angular velocity, or even angular acceleration if needed. It helps us understand not just how far, but in what manner, an object has rotated.
Kinematic Equations for Rotation
Kinematic equations for rotation help relate different aspects of rotational motion. Just like with linear motion, they help us solve for unknowns using known quantities. These equations involve angular displacement, angular velocity, angular acceleration, and time.

For the exercise involving the basketball player, we used the equation: \[ \theta = \frac{1}{2} (\omega_i + \omega_f) \cdot t \]where:
  • \( \theta \) is the angular displacement
  • \( \omega_i \) is the initial angular velocity
  • \( \omega_f \) is the final angular velocity
  • \( t \) is the time
This particular equation doesn't involve angular acceleration explicitly, making it perfect for situations where you aren't given or required to find it.

Understanding these equations equips you to analyze and solve various problems regarding rotational motion, much like you would with everyday linear motion questions.

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Most popular questions from this chapter

In general, does the average angular acceleration of a rotating object have the same direction as its initial angular velocity \(\omega_{0}\), its final angular velocity \(\omega\), or the difference \(\omega-\omega_{0}\) between its final and initial angular velocities? (b) The table that follows lists four pairs of initial and final angular velocities for a rotating fan blade. Determine the direction (positive or negative) of the average angular acceleration for each pair. Provide reasons for your answers. $$ \begin{array}{|c|c|c|} \hline & \text { Initial angular velocity } \omega_{0} & \text { Final angular velocity } \omega \\ \hline \text { (a) } & +2.0 \mathrm{rad} / \mathrm{s} & +5.0 \mathrm{rad} / \mathrm{s} \\ \hline \text { (b) } & +5.0 \mathrm{rad} / \mathrm{s} & +2.0 \mathrm{rad} / \mathrm{s} \\ \hline \text { (c) } & -7.0 \mathrm{rad} / \mathrm{s} & -3.0 \mathrm{rad} / \mathrm{s} \\ \hline \text { (d) } & +4.0 \mathrm{rad} / \mathrm{s} & -4.0 \mathrm{rad} / \mathrm{s} \\ \hline \end{array} $$ Problem The elapsed time for each of the four pairs of angular velocities is \(4.0 \mathrm{~s}\). Find the average angular acceleration (magnitude and direction) for each of the four pairs. Be sure that your directions agree with those found in the Concept Question. Concept Question In the table are listed the initial angular velocity \(\omega_{0}\) and the angular acceleration \(\alpha\) of four rotating objects at a given instant in time.

Interactive Solution \(\underline{8.23}\) at offers a model for this problem. The drive propeller of a ship starts from rest and accelerates at \(2.90 \times 10^{-3} \mathrm{rad} / \mathrm{s}^{2}\) for \(2.10 \times 10^{3} \mathrm{~s}\). For the next \(1.40 \times 10^{3} \mathrm{~s}\) the propeller rotates at a constant angular speed. Then it decelerates at \(2.30 \times 10^{-3} \mathrm{rad} / \mathrm{s}^{2}\) until it slows (without reversing direction) to an angular speed of 4.00 \(\mathrm{rad} / \mathrm{s}\). Find the total angular displacement of the propeller.

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 \(\mathrm{rad} / \mathrm{s}^{2}\)

An electric fan is running on HIGH. After the LOW button is pressed, the angular speed of the fan decreases to \(83.8 \mathrm{rad} / \mathrm{s}\) in \(1.75 \mathrm{~s}\). The deceleration is \(42.0 \mathrm{rad} / \mathrm{s}^{2}\). Determine the initial angular speed of the fan.

A person is riding a bicycle, and its wheels have an angular velocity of \(+20.0 \mathrm{rad} / \mathrm{s}\). Then, the brakes are applied and the bike is brought to a uniform stop. During braking, the angular displacement of each wheel is +15.92 revolutions. (a) How much time does it take for the bike to come to rest? (b) What is the angular acceleration (in \(\mathrm{rad} / \mathrm{s}^{2}\) ) of each wheel?
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