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Chapter 8: Problem 72

After 10.0 s, a spinning roulette wheel at a casino has slowed down to an angular velocity of \(+1.88 \mathrm{rad} / \mathrm{s} .\) During this time, the wheel has an angular acceleration of \(-5.04 \mathrm{rad} / \mathrm{s}^{2} .\) Determine the angular displacement of the wheel.

Short Answer

Expert verified
The angular displacement of the wheel is 270.8 radians.

Step by step solution

01

Identify the given values

We are given:- Final angular velocity, \( \omega_f = +1.88 \; \text{rad/s} \)- Angular acceleration, \( \alpha = -5.04 \; \text{rad/s}^2 \)- Time, \( t = 10.0 \; \text{s} \)We need to find the initial angular velocity, \( \omega_i \), before calculating the angular displacement.
02

Use the formula for final angular velocity

The formula that relates initial angular velocity (\( \omega_i \)), final angular velocity (\( \omega_f \)), angular acceleration (\( \alpha \)), and time (\( t \)) is:\[ \omega_f = \omega_i + \alpha t \]Substitute the given values into the equation:\[ 1.88 = \omega_i + (-5.04) \times 10 \]
03

Solve for initial angular velocity

Rearrange the equation to solve for \( \omega_i \):\[ \omega_i = 1.88 + 5.04 \times 10 \]\[ \omega_i = 1.88 + 50.4 \]\[ \omega_i = 52.28 \; \text{rad/s} \]
04

Use the angular displacement formula

Now that we have \( \omega_i \), use the angular displacement formula:\[ \theta = \omega_i t + \frac{1}{2} \alpha t^2 \]Substitute the known values:\[ \theta = 52.28 \times 10 + \frac{1}{2} (-5.04) \times (10)^2 \]
05

Calculate the angular displacement

Carry out the calculations:\[ \theta = 522.8 + \frac{1}{2} \times (-5.04) \times 100 \]\[ \theta = 522.8 - 252 \]\[ \theta = 270.8 \; \text{radians} \]
06

Conclusion: Final result

The angular displacement of the wheel after 10 seconds is \( 270.8 \; \text{radians} \).

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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 understanding how an object rotates or spins around a fixed point or axis. It refers to the rate at which an object, say a roulette wheel, spins. In simple terms, it measures how fast an angle changes with respect to time.
For example, if a wheel completes one full spin, which is equal to a 360-degree rotation or 2蟺 radians, in one second, the angular velocity is 2蟺 radians per second. In your exercise, the wheel has slowed down to an angular velocity of +1.88 radians per second after 10 seconds. This positive sign means the direction of the rotation hasn't changed, though the wheel is slowing.
  • Angular velocity is measured in radians per second (rad/s).
  • It is a vector quantity, meaning it has both magnitude and direction.
Understanding angular velocity helps you predict how fast or slow a rotational movement is at any given moment.
Angular Acceleration
Angular acceleration is crucial for determining how quickly the rotational speed of an object is changing. It's similar to linear acceleration, but it deals with rotation instead of straight-line motion. When a wheel speeds up or slows down, angular acceleration is at play.
In the problem, the wheel experiences an angular acceleration of -5.04 rad/s虏 over 10 seconds. The negative sign indicates that the wheel is decelerating, or slowing its rate of spin.
  • Angular acceleration is measured in radians per second squared (rad/s虏).
  • It describes how the angular velocity changes over time.
This concept helps in calculating how much time it would take for the wheel to reach a certain speed or come to a halt, which is useful in the field of dynamics.
Kinematics
Kinematics is the branch of mechanics that deals with motion without considering the forces that cause it. When applied to rotational motion, kinematics focuses on relationships between angular displacement, velocity, acceleration, and time.
The steps you followed in solving the exercise鈥攃alculating the initial angular velocity and then using it to find angular displacement鈥攁re part of rotational kinematics.
  • Kinematic equations, like those used in the solution, help link various aspects of motion.
  • They provide a systematic approach to solve complex problems related to rotational dynamics.
Understanding kinematic equations lets you analyze motions that involve rotations, like how far a wheel travels or how long it takes to reach a certain speed.
Rotational Motion
Rotational motion involves objects moving around a central point or axis, like the spinning roulette wheel in the exercise. This type of motion is all about angles and how they change over time, making it distinct from linear motion, which concerns straight paths.
For the wheel, its path and speed changes over time determine its rotational motion characteristics. The final solution using angular displacement shows how far the wheel has turned.
  • It involves concepts such as moment of inertia and torque, but your problem focuses on angular displacement as the key outcome.
  • Rotational motion is common in everyday life, from celestial motions to everyday appliances.
Grasping these ideas opens doors to further understanding dynamics and complex systems in physics, engineering, and other scientific fields.

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Most popular questions from this chapter

The drawing shows a device that can be used to measure the speed of a bullet. The device consists of two rotating disks, separated by a distance of \(d=0.850 \mathrm{m},\) and rotating with an angular speed of \(95.0 \mathrm{rad} / \mathrm{s} .\) The bullet first passes through the left disk and then through the right disk. It is found that the angular displacement between the two bullet holes is \(\theta=0.240\) rad. From these data, determine the speed of the bullet.

A dentist causes the bit of a high-speed drill to accelerate from an angular speed of \(1.05 \times 10^{4} \mathrm{rad} / \mathrm{s}\) to an angular speed of \(3.14 \times 10^{4} \mathrm{rad} / \mathrm{s}\) In the process, the bit turns through \(1.88 \times 10^{4}\) rad. Assuming a constant angular acceleration, how long would it take the bit to reach its maximum speed of \(7.85 \times 10^{4} \mathrm{rad} / \mathrm{s},\) starting from rest?

A rectangular plate is rotating with a constant angular speed about an axis that passes perpendicularly through one corner, as the drawing shows. The centripetal acceleration measured at corner \(A\) is \(n\) times as great as that measured at corner \(B\). What is the ratio \(L_{1} / L_{2}\) of the lengths of the sides of the rectangle when \(n=2.00 ?\)

A spinning wheel on a fireworks display is initially rotating in a counterclockwise direction. The wheel has an angular acceleration of -4.00 \(\mathrm{rad} / \mathrm{s}^{2} .\) Because of this acceleration, the angular velocity of the wheel changes from its initial value to a final value of \(-25.0 \mathrm{rad} / \mathrm{s} .\) While this change occurs, the angular displacement of the wheel is zero. (Note the similarity to that of a ball being thrown vertically upward, coming to a momentary halt, and then falling downward to its initial position.) Find the time required for the change in the angular velocity to occur.

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.
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