Question

A bicycle wheel has an initial angular velocity of 1.50 rad/s. (a) If its angular acceleration is constant and equal to 0.200 rad/s\(^2\), what is its angular velocity at \(t =\) 2.50 s? (b) Through what angle has the wheel turned between \(t =\) 0 and \(t =\) 2.50 s?

Short answer

(a) 2.00 rad/s, (b) 4.375 rad

Step-by-step solution

Identify Given Values

From the problem, we have:- Initial angular velocity, \( \omega_0 = 1.50 \, \text{rad/s} \)- Angular acceleration, \( \alpha = 0.200 \, \text{rad/s}^2 \)- Time, \( t = 2.50 \, \text{s} \)

Find Final Angular Velocity

We use the equation for angular velocity with constant angular acceleration:\[ \omega = \omega_0 + \alpha t \]Substitute the given values:\[ \omega = 1.50 \, \text{rad/s} + (0.200 \, \text{rad/s}^2)(2.50 \, \text{s}) \]\[ \omega = 1.50 \, \text{rad/s} + 0.50 \, \text{rad/s} \]\[ \omega = 2.00 \, \text{rad/s} \]Thus, the final angular velocity at \( t = 2.50 \, \text{s} \) is 2.00 rad/s.

Calculate Total Angular Displacement

Use the angular displacement formula:\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]Substitute the known values:\[ \theta = (1.50 \, \text{rad/s})(2.50 \, \text{s}) + \frac{1}{2}(0.200 \, \text{rad/s}^2)(2.50 \, \text{s})^2 \]\[ \theta = 3.75 \, \text{rad} + \frac{1}{2}(0.200)(6.25) \, \text{rad} \]\[ \theta = 3.75 \, \text{rad} + 0.625 \, \text{rad} \]\[ \theta = 4.375 \, \text{rad} \]Thus, the total angle through which the wheel has turned is 4.375 radians.

Key concepts

Angular Velocity

Angular velocity is a measure of how quickly an object rotates or spins around a central point or axis. In this context of the bicycle wheel, it describes how fast the wheel is spinning. It is a vector quantity, meaning it has both a magnitude and a direction, but in most simple scenarios such as a bicycle wheel, we focus primarily on the magnitude.
Angular velocity is often represented by the Greek letter omega (\( \omega \)), and it is measured in radians per second (rad/s). Imagine it like the speed of a car, but for a rotating object instead. When a wheel spins faster, its angular velocity increases.

To calculate angular velocity when angular acceleration is constant, you use this equation:
\[ \omega = \omega_0 + \alpha t \]
Where:

• \( \omega_0 \) is the initial angular velocity (how fast it was spinning to begin with)
• \( \alpha \) is the angular acceleration
• \( t \) is the time elapsed

In the original problem, when the bicycle wheel started with an angular velocity of 1.50 rad/s and an angular acceleration of 0.200 rad/s² was applied for 2.5 seconds, its final angular velocity increased to 2.00 rad/s as calculated using the formula. This indicates the wheel spins faster over time due to the constant acceleration.

Angular Acceleration

Angular acceleration is the rate at which angular velocity changes with time. It tells us how quickly the angular speed of a spinning object, like a wheel, is increasing or decreasing. It is represented by the Greek letter alpha (\( \alpha \)) and measured in radians per second squared (rad/s²).

Think of it as how hard you're pushing or pulling on a spinning wheel to change its speed. If a wheel is accelerating, it means the angular velocity of the wheel is either increasing or decreasing.

In our exercise, the bicycle wheel has a constant angular acceleration of 0.200 rad/s². This means every second, the speed at which the wheel spins increases by 0.200 rad/s. This constant acceleration is crucial when using the formula for finding the change in angular velocity over time, as it simplifies the calculation by allowing us to assume a linear increase. Without constant acceleration, the equations would be more complex, involving calculus to account for variable changes over time.

Angular Displacement

Angular displacement is the measure of how much an object has rotated or turned, generally represented by the Greek letter theta (\( \theta \)). It’s akin to distance in linear motion but deals with angles in rotational motion.
Measured in radians, it gives us the total change in the angle of the rotating object from start to finish.

To calculate angular displacement, especially when dealing with constant angular acceleration, you can use the formula:
\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]
Where:

• \( \omega_0 \) is the initial angular velocity
• \( \alpha \) is the angular acceleration
• \( t \) is time

Using this formula, you start with the initial rotation and add the effect of acceleration over a specified time.
In this problem, with the given values, the total amount that the wheel rotates in 2.5 seconds comes out to be 4.375 radians. This tells you exactly how far the wheel has turned from its initial position.