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. What is the angular displacement of the wheel at t = 7.00 s?

Verified step by step guidance

1. Determine the angular acceleration (α) of the wheel. Since the angular velocity (ω_z) changes linearly with time, the angular acceleration is constant. Use the formula: $\alpha =\frac{\omega }{}$_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 for angular acceleration: $\omega$_f = +4.00 rad/s, $\omega$_i = -6.00 rad/s, and $t=\; 7.00$ s. This will give you the value of $\alpha$.
3. Use the kinematic equation for angular displacement: $\theta =\omega$_it + 12αt². Here, $\theta$ is the angular displacement, $\omega$_i is the initial angular velocity, $t$ is the time, and $\alpha$ is the angular acceleration.
4. Substitute the known values into the angular displacement equation: $\omega$_i = -6.00 rad/s, $t=\; 7.00$ s, and the calculated value of $\alpha$ from step 2. Simplify the terms to find $\theta$.
5. Interpret the result. The sign of the angular displacement will indicate the direction of rotation (positive for counterclockwise and negative for clockwise). Ensure the units of the final angular displacement are in radians.

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 Displacement

Angular displacement refers to the angle through which an object has rotated about a specific axis in a given time period. It is typically measured in radians and can be calculated by integrating angular velocity over time. In this scenario, the angular displacement can be determined by finding the area under the angular velocity-time graph from t = 0 to t = 7.00 s.

Kinematics of Rotational Motion

The kinematics of rotational motion describes the relationships between angular displacement, angular velocity, and angular acceleration. When angular velocity changes linearly, as in this case, the angular acceleration can be calculated as the change in angular velocity divided by the time interval. This concept is essential for determining the total angular displacement when the angular velocity is not constant.