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Chapter 8: Problem 89

Suppose a roulette wheel is spinning at \(1 \mathrm{rev} / \mathrm{s}\). How long will it take for the wheel to come to rest if it experiences an angular acceleration of \(-0.02 \mathrm{rad} / \mathrm{s}^{2}\) ? How many rotations will it complete in that time?

Short Answer

Expert verified
The length of time it will take for the wheel to come to rest and the total number of rotations it makes in that time will be determined by the solutions obtained from steps 2 and 3.

Step by step solution

01

Convert angular velocity to SI units

The angular velocity is given as \(1 \ rev/sec\). One complete revolution is equivalent to \(2\pi\) radians. Therefore, the angular velocity in radians per second can be calculated as follows: \( \omega_{i} = 1\ rev/sec \times 2\pi \ rad/rev = 2\pi \ rad/sec\).
02

Calculate the time to rest

The time needed for the wheel to come to rest can be calculated using the equation \( \omega_{f} = \omega_{i} + \alpha t\). As the wheel is coming to rest, \( \omega_f = 0 \ rad/s\). Solving for time, we get: \( t = (\omega_{f} - \omega_{i}) / \alpha = (0 - 2\pi) / -0.02 \).
03

Calculate the number of completed rotations

The total number of rotations the wheel completes before it comes to rest can be determined from the angular displacement. The displacement can be found using the equation: \( \theta = \omega_{i} \cdot t + 0.5 \cdot \alpha \cdot t^{2}\). After finding the displacement in radians, convert this into the number of rotations by dividing by \(2\pi\).

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Key Concepts

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

Angular Velocity
Angular velocity describes how fast something is spinning. In the exercise, the roulette wheel spins at \(1 \text{ rev/s}\). This already gives us a clue about its speed. To work with equations, we often need to convert this into radians per second, the standard unit. Since one full revolution equals \(2\pi\) radians, multiplying \(1 \text{ rev/s}\) by \(2\pi\) gives \(2\pi \text{ rad/s}\).
This conversion is crucial for calculating time, distance, and other related rotational properties.
Angular Acceleration
Angular acceleration is about how fast an object's spin changes. In our problem, the angular acceleration is \(-0.02 \text{ rad/s}^2\). The negative sign indicates deceleration or slowing down.
The unit, radian per second squared, measures how much the angular velocity changes per second per second. This concept helps us understand how long it takes the roulette wheel to stop spinning. We use angular acceleration in kinematic equations to figure out time and rotation details.
Rotational Motion
Rotational motion occurs when an object spins around an axis. In this case, the axis is the center of the roulette wheel. Think of how easily it keeps spinning smoothly or slows down steadily.
We can analyze rotational motion using angular velocity and acceleration. These help us determine how long an object moves and how much it turns. By linking together these key ideas, we paint a fuller picture of how objects twist and turn.
SI Units
The International System of Units (SI) standardizes measurements to ensure consistency. For our calculations, we primarily relied on radians for angular measures and seconds for time.
  • Angular velocity typically uses rad/s, showing the spin rate in radians per second.
  • Angular acceleration employs rad/s虏, illustrating change per second every second.
These units simplify equations and calculations, helping us communicate clear solutions and results without confusion.
Kinematics
Kinematics focuses on motion, covering both linear and rotational forms. It explains how objects move without digging into forces causing these motions.
In rotational kinematics, we explore how spinning objects like wheels and tops behave over time. By using formulas for angular velocity, acceleration, and displacement, we solve for various quantities. These concepts play a big role in helping us predict movement patterns, whether they be straightforward translations or intricate spins.

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