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. During what time interval is the speed of the wheel increasing? Decreasing?

Verified step by step guidance

1. Step 1: Understand the problem. The angular velocity ω_z changes linearly with time, meaning it follows a straight-line relationship. The angular velocity starts at -6.00 rad/s at t = 0 and ends at +4.00 rad/s at t = 7.00 s. Counterclockwise rotation is positive, and we need to determine the intervals during which the speed (magnitude of angular velocity) is increasing and decreasing.
2. Step 2: Write the equation for angular velocity as a function of time. Since ω_z changes linearly, it can be expressed as ω_z = m * t + b, where m is the slope (rate of change of angular velocity) and b is the initial angular velocity. Calculate the slope m using the formula m = (ω_z_final - ω_z_initial) / (t_final - t_initial).
3. Step 3: Determine the time at which the angular velocity crosses zero. The speed of the wheel is the magnitude of ω_z, |ω_z|. The speed increases when the magnitude of ω_z moves away from zero and decreases when it moves toward zero. Solve for t when ω_z = 0 using the linear equation derived in Step 2.
4. Step 4: Analyze the intervals. From t = 0 to the time when ω_z = 0, the angular velocity is negative and moving toward zero, meaning the speed is decreasing. From the time when ω_z = 0 to t = 7.00 s, the angular velocity is positive and moving away from zero, meaning the speed is increasing.
5. Step 5: Summarize the intervals. The speed of the wheel is decreasing during the interval from t = 0 to the time when ω_z = 0, and increasing during the interval from the time when ω_z = 0 to t = 7.00 s. Use the calculated time from Step 3 to specify these intervals.

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 rotation. In this question, the angular velocity changes from -6.00 rad/s to +4.00 rad/s, indicating a transition from clockwise to counterclockwise rotation.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, typically expressed in radians per second squared (rad/s²). It determines whether the speed of rotation is increasing or decreasing. In this scenario, since the angular velocity is increasing from a negative to a positive value, the angular acceleration is positive, indicating that the wheel's speed is increasing during this time.

Direction of Rotation

The direction of rotation is crucial in determining whether the speed of the wheel is increasing or decreasing. In this problem, counterclockwise rotation is defined as positive, while clockwise is negative. The change in angular velocity from negative to positive signifies a shift in direction, which affects the interpretation of speed changes during the specified time interval.