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. (a) Is the angular acceleration during this time interval positive or negative?

Verified step by step guidance

1. Identify the initial angular velocity (\(\omega_{z_i}\)) and the final angular velocity (\(\omega_{z_f}\)) from the problem statement. Here, \(\omega_{z_i} = -6.00 \, \text{rad/s}\) and \(\omega_{z_f} = +4.00 \, \text{rad/s}\).
2. Note the time interval during which the angular velocity changes. It starts at \(t_i = 0 \, \text{s}\) and ends at \(t_f = 7.00 \, \text{s}\).
3. Calculate the change in angular velocity (\(\Delta \omega_z\)) by subtracting the initial angular velocity from the final angular velocity, i.e., \(\Delta \omega_z = \omega_{z_f} - \omega_{z_i}\).
4. Calculate the time duration (\(\Delta t\)) over which this change occurs, i.e., \(\Delta t = t_f - t_i\).
5. Determine the angular acceleration (\(\alpha_z\)) using the formula \(\alpha_z = \frac{\Delta \omega_z}{\Delta t}\). The sign of \(\alpha_z\) will indicate whether the angular acceleration is positive or negative.

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 direction and magnitude over time.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, typically measured in radians per second squared (rad/s²). It can be positive or negative, depending on whether the angular velocity is increasing or decreasing. In this scenario, determining the sign of angular acceleration involves analyzing the change in angular velocity from negative to positive over the specified time interval.

Linear Change in Angular Velocity

When angular velocity changes linearly with time, it implies a constant angular acceleration. This means that the angular velocity increases or decreases at a steady rate. In the given problem, the angular velocity transitions from -6.00 rad/s to +4.00 rad/s over 7 seconds, allowing us to calculate the angular acceleration as the difference in angular velocity divided by the time interval.