CALC A fan blade rotates with angular velocity given by ω_z(t) = g - bt^2, where g = 5.00 rad/s and b = 0.800 rad/s^3. (a) Calculate the angular acceleration as a function of time.

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1. Identify the given function for angular velocity, \(\omega_z(t) = g - bt^2\), where \(g = 5.00 \, \text{rad/s}\) and \(b = 0.800 \, \text{rad/s}^3\).
2. Recall that angular acceleration, \(\alpha\), is the time derivative of angular velocity, \(\omega\). Therefore, calculate \(\alpha\) by differentiating \(\omega_z(t)\) with respect to time \(t\).
3. Differentiate the function \(\omega_z(t) = g - bt^2\). The derivative of a constant \(g\) is 0, and the derivative of \(-bt^2\) with respect to \(t\) is \(-2bt\).
4. Combine the results of the differentiation to express the angular acceleration as a function of time: \(\alpha(t) = -2bt\).
5. Substitute the value of \(b\) into the expression for \(\alpha(t)\) to get the final expression for angular acceleration as a function of time: \(\alpha(t) = -2 \times 0.800 \, \text{rad/s}^3 \times t\).

Key Concepts

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, expressed in radians per second (rad/s). In this context, the angular velocity of the fan blade is given as a function of time, indicating that it changes as the blade rotates. Understanding this concept is crucial for analyzing the motion of rotating objects.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, typically expressed in radians per second squared (rad/s²). It can be calculated by taking the derivative of the angular velocity function with respect to time. In this problem, finding the angular acceleration involves differentiating the given function ω_z(t) to understand how the rotation speed of the fan blade changes over time.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the rate at which a quantity changes. In physics, it is often used to derive quantities like velocity from position or acceleration from velocity. For this question, applying differentiation to the angular velocity function will yield the angular acceleration, which is essential for understanding the dynamics of the rotating fan blade.