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A rider on a mountain bike is traveling to the left in the figure. Each wheel has an angular velocity of \(+21.7 \mathrm{rad} / \mathrm{s},\) where, as usual, the plus sign indicates that the wheel is rotating in the counterclockwise direction. (a) To pass another cyclist, the rider pumps harder, and the angular velocity of the wheels increases from \(+21.7 \mathrm{to}+28.5 \mathrm{rad} / \mathrm{s}\) in a time of \(3.50 \mathrm{s}\) (b) After passing the cyclist, the rider begins to coast, and the angular velocity of the wheels decreases from \(+28.5 \mathrm{to}+15.3 \mathrm{rad} / \mathrm{s}\) in a time of \(10.7 \mathrm{s}\) Concepts: (i) Is the angular acceleration positive or negative when the rider is passing the cyclist and the angular speed of the wheels is increasing? (ii) Is the angular acceleration positive or negative when the rider is coasting and the angular speed of the wheels is decreasing? Calculations: In both instances, (a) and (b), determine the magnitude and direction of the angular acceleration (assumed constant) of the wheels.

Step by step solution

01

Determine the Initial and Final Angular Velocities for Part (a)

For part (a), the initial angular velocity \(\omega_i\) is \(+21.7 \ \mathrm{rad/s}\) and the final angular velocity \(\omega_f\) is \(+28.5 \ \mathrm{rad/s}\). The time interval \(\Delta t\) is \(3.50 \ \mathrm{s}\). We will use these values to find the angular acceleration.

02

Calculate Angular Acceleration for Part (a)

Angular acceleration \(\alpha\) is found using the formula \(\alpha = \frac{\omega_f - \omega_i}{\Delta t}\). Substituting the known values, \[\alpha = \frac{28.5 \ \mathrm{rad/s} - 21.7 \ \mathrm{rad/s}}{3.50 \ \mathrm{s}} = \frac{6.8 \ \mathrm{rad/s}}{3.50 \ \mathrm{s}} \approx 1.94 \ \mathrm{rad/s^2}.\] Hence, the angular acceleration is \(+1.94 \ \mathrm{rad/s^2}\) and positive, indicating an increase in speed.

03

Determine the Initial and Final Angular Velocities for Part (b)

For part (b), the initial angular velocity \(\omega_i\) is \(+28.5 \ \mathrm{rad/s}\) and the final angular velocity \(\omega_f\) is \(+15.3 \ \mathrm{rad/s}\). The time interval \(\Delta t\) is \(10.7 \ \mathrm{s}\). We will use these values to find the angular acceleration.

04

Calculate Angular Acceleration for Part (b)

Angular acceleration \(\alpha\) is found using the formula \(\alpha = \frac{\omega_f - \omega_i}{\Delta t}\). Substituting the known values, \[\alpha = \frac{15.3 \ \mathrm{rad/s} - 28.5 \ \mathrm{rad/s}}{10.7 \ \mathrm{s}} = \frac{-13.2 \ \mathrm{rad/s}}{10.7 \ \mathrm{s}} \approx -1.23 \ \mathrm{rad/s^2}.\] Therefore, the angular acceleration is \(-1.23 \ \mathrm{rad/s^2}\) and negative, indicating a decrease in speed.

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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 measure of how fast an object rotates or spins around an axis. It is typically denoted by the Greek letter \( \omega \). For a rotating wheel, angular velocity is measured in radians per second (rad/s), which describes the change in angular displacement over time. Consider a mountain bike wheel as an example. If the wheel rotates counterclockwise, its angular velocity is positive. This is because, by convention, counterclockwise rotation is considered positive.

• Initial angular velocity: The speed at which the wheel started its rotation.
• Final angular velocity: The speed at which the wheel ends its rotation for a given time interval.

When analyzing problems involving angular velocity, it's important to identify both the initial and final angular velocities so that changes over time can be calculated effectively.

Constant Angular Acceleration

Constant angular acceleration occurs when the rate of change of angular velocity is uniform over a period of time. In simpler terms, it's when an object experiencing rotation changes its speed at a steady rate. Angular acceleration (denoted as \( \alpha \)) is measured in radians per second squared (rad/s²). It can be calculated using the formula: \[ \alpha = \frac{\omega_f - \omega_i}{\Delta t} \]where \( \omega_f \) is the final angular velocity, \( \omega_i \) is the initial angular velocity, and \( \Delta t \) is the time interval over which the change occurs.

• A positive angular acceleration implies that the angular speed is increasing.
• A negative angular acceleration means an object is slowing down in its rotation.

In our bike wheel example, we noticed positive angular acceleration when the rider increased speed. Conversely, when slowing down, the angular acceleration became negative.

Angular Speed

Angular speed refers to the magnitude of the angular velocity, essentially the rate at which the wheel revolves around its axis regardless of direction. It shares the same unit as angular velocity, namely radians per second (rad/s), but it doesn't consider the direction, merely how fast the rotation occurs.
While angular velocity can be negative or positive (indicating direction), angular speed is always non-negative, representing only the size of the velocity vector.

• Increases when a wheel accelerates (positive acceleration).
• Decreases when a wheel slows down (negative acceleration).

Understanding angular speed is crucial when determining how fast a rotational motion is without focusing on the rotational direction. It's especially useful in contexts like the exercise presented, where wheel speed alters as the rider accelerates and then decelerates.