A wheel is rotating about an axis that is in the z-direction. The angular velocity ω_z is -6.00 rad/s at t = 0, increases linearly with time, and is +4.00 rad/s at t = 7.00 s. We have taken counterclockwise rotation to be positive. (c) What is the angular displacement of the wheel at t = 7.00 s?

Verified step by step guidance

1. Determine the angular acceleration (\(\alpha\)) by using the formula for angular acceleration, which is the change in angular velocity (\(\Delta \omega\)) divided by the change in time (\(\Delta t\)). The formula is \(\alpha = \frac{\Delta \omega}{\Delta t}\).
2. Calculate the change in angular velocity (\(\Delta \omega\)) by subtracting the initial angular velocity (\(\omega_{z, initial}\)) from the final angular velocity (\(\omega_{z, final}\)).
3. Substitute the values into the angular acceleration formula to find \(\alpha\).
4. Use the formula for angular displacement (\(\theta\)) when angular acceleration is constant: \(\theta = \omega_{z, initial} \cdot t + \frac{1}{2} \cdot \alpha \cdot t^2\).
5. Substitute the values of \(\omega_{z, initial}\), \(\alpha\), and \(t\) into the angular displacement formula to find the angular displacement at \(t = 7.00 s\).

Key Concepts

Angular Velocity

Angular velocity is a vector quantity that represents the rate of rotation of an object around an axis. It is measured in radians per second (rad/s) and indicates both the speed and direction of the rotation. In this question, the angular velocity changes from -6.00 rad/s to +4.00 rad/s, indicating a change in the direction of rotation over time.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, typically expressed in radians per second squared (rad/s²). In this scenario, the angular velocity increases linearly, suggesting a constant angular acceleration. This concept is crucial for determining how the angular velocity changes from its initial to final value over the given time interval.

Angular Displacement

Angular displacement refers to the angle through which an object has rotated about a specific axis in a given time period, measured in radians. It can be calculated using the initial and final angular velocities and the time interval, often using the formula θ = ω_initial * t + 0.5 * α * t², where α is the angular acceleration. This concept is essential for finding the total rotation of the wheel at t = 7.00 s.