Question

As you start riding a bicycle, the wheels begin at rest and have an angular acceleration of \(2.3 \mathrm{rad} / \mathrm{s}^{2}\). What is the angular speed of the wheels after \(3.8 \mathrm{~s}\) ?

Short answer

The angular speed after 3.8 seconds is 8.74 rad/s.

Step-by-step solution

Understand the Given Data

We're given the angular acceleration of the wheels, \( \alpha = 2.3 \, \text{rad/s}^2 \), and the time duration, \( t = 3.8 \, \text{s} \). The initial angular speed is \( \omega_0 = 0 \, \text{rad/s} \) because the wheels start from rest. We need to find the final angular speed, \( \omega \).

Use the Angular Motion Equation

To find the final angular speed, use the equation for angular motion: \( \omega = \omega_0 + \alpha t \). Substitute the given values into this equation.

Substitute the Known Values

Substitute \( \omega_0 = 0 \, \text{rad/s} \), \( \alpha = 2.3 \, \text{rad/s}^2 \), and \( t = 3.8 \, \text{s} \) into the equation: \[ \omega = 0 + 2.3 \, \text{rad/s}^2 \times 3.8 \, \text{s} \].

Calculate the Angular Speed

Perform the multiplication to find \( \omega \): \( \omega = 2.3 \, \text{rad/s}^2 \times 3.8 \, \text{s} = 8.74 \, \text{rad/s} \).

Conclusion

The angular speed of the bicycle wheels after 3.8 seconds is \( 8.74 \, \text{rad/s} \).

Key concepts

Angular Acceleration

Angular acceleration is a key concept in the study of rotational motion. Angular acceleration refers to how quickly the angular speed of an object changes over time. It is denoted by the symbol \( \alpha \) and measured in radians per second squared \( \text{rad/s}^2 \). This physical quantity plays a role similar to linear acceleration but in rotational systems. To understand it better, think of it as the rate of change of the angular velocity. In our bicycle example, as the wheels spin faster, they experience an angular acceleration of \(2.3 \, \text{rad/s}^2\). This means that with each passing second, the speed at which the wheels make a complete turn is increasing by \(2.3 \, \text{rad/s}^2\).Angular acceleration can be either positive or negative.

• A positive angular acceleration indicates that the object is speeding up.
• A negative angular acceleration signifies that the object is slowing down.

In our problem, the wheels start from rest, so the acceleration eventually leads to an increase in their angular speed.

Angular Speed

Angular speed is the rate at which an object rotates or revolves around another point. It measures how much angle an object covers in a specific duration and is denoted by the symbol \( \omega \). Angular speed is measured in radians per second \( \text{rad/s} \).In simple terms, angular speed tells us how fast an object is rotating. For the bicycle problem, once we know the angular acceleration and the time elapsed, we can calculate the angular speed using the equation for angular motion:\[ \omega = \omega_0 + \alpha t \]Where

• \( \omega_0 \) is the initial angular speed (which is zero since the wheels start from rest),
• \( \alpha \) is the angular acceleration, and
• \( t \) is the time duration.

With this formula, we find that after \(3.8 \text{ s}\), the wheels achieve an angular speed of \(8.74 \, \text{rad/s}\). If the angular speed is high, the wheels rotate swiftly, indicating brisk motion.

Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the forces causing the motion. In the context of rotational motion, we use angular kinematics to analyze movements involving rotation.Angular motion shares many principles with linear motion, but uses slightly different terminology:

• Position is measured in angles, usually in radians.
• Velocity is termed as angular speed \( \omega \).
• Acceleration is known as angular acceleration \( \alpha \).

The equations of kinematics provide a framework for understanding rotational motion. Using the equation \( \omega = \omega_0 + \alpha t \) from angular kinematics, we link the initial angular speed, angular acceleration, and time to find the final angular speed.These equations are crucial for analyzing various physical systems like wheels, gears, and disks, which rotate rather than move in a straight line. As such, mastering kinematics is fundamental for predicting the outcome of rotational motions.

Rotational Motion

Rotational motion involves the motion of objects that rotate about an axis. This form of motion is prevalent in numerous systems around us, from planets moving around the sun to electrons orbiting the nucleus and everyday items like fans and wheels spinning. When dealing with rotational motion,

• angular displacement measures the change in position around a circle,
• angular speed quantifies how fast an object is rotating, and
• angular acceleration indicates whether the rotation is speeding up or slowing down.

Rotational motion is governed by similar principles as linear motion, such as Newton's laws, which can be applied with slight modifications. The central concept is understanding the dynamic behavior of rotating objects, such as calculating angular velocities or accelerations, like in our bicycle problem. This knowledge is essential not only for academic purposes but also for practical applications in engineering, mechanics, and various technological fields where rotational systems are foundational. Mastering rotational motion concepts helps in solving real-world problems and designing efficient mechanical systems.