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

Verified step by step guidance

1. Determine the formula for angular acceleration (α). Angular acceleration is defined as the rate of change of angular velocity (ω) with respect to time (t). The formula is: $\alpha =\frac{\omega }{}$_f - ω_it, where $\omega$_f is the final angular velocity, $\omega$_i is the initial angular velocity, and $t$ is the time interval.
2. Substitute the given values into the formula. The initial angular velocity is $\omega$_i = -6.00 rad/s, the final angular velocity is $\omega$_f = +4.00 rad/s, and the time interval is $t=\; 7.00$ s.
3. Calculate the numerator of the formula, which is the change in angular velocity: $\omega$_f - ω_i = 4.00 - (-6.00). Simplify this expression to find the total change in angular velocity.
4. Divide the change in angular velocity by the time interval to find the angular acceleration: $\alpha =\frac{10.00}{7.00}$. This will give the angular acceleration in rad/s².
5. Interpret the sign of the angular acceleration. Since the angular velocity is increasing from a negative value to a positive value, the angular acceleration is positive. This indicates that the wheel is speeding up in the counterclockwise direction.

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 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, the angular acceleration can be determined by analyzing the change in angular velocity from -6.00 rad/s to +4.00 rad/s over the time interval of 7 seconds.

Direction of Rotation

The direction of rotation is crucial in determining the sign of angular velocity and angular acceleration. In this problem, counterclockwise rotation is defined as positive, while clockwise rotation is negative. The change in angular velocity from a negative to a positive value indicates a transition from clockwise to counterclockwise rotation, which implies that the angular acceleration must be positive during the time interval considered.