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.

Verified step by step guidance

1. Understand that angular acceleration is the derivative of angular velocity with respect to time. In this problem, the angular velocity ω_z(t) is given as a function of time: ω_z(t) = g - bt^2.
2. Identify the constants in the equation: g = 5.00 rad/s and b = 0.800 rad/s^3.
3. To find the angular acceleration α(t), take the derivative of ω_z(t) with respect to time t. The derivative of a constant g is 0, and the derivative of -bt^2 is -2bt.
4. Apply the derivative: α(t) = d(ω_z(t))/dt = 0 - 2bt = -2bt.
5. Substitute the value of b into the expression for angular acceleration: α(t) = -2 * 0.800 * t = -1.60t rad/s^2. This gives the angular acceleration as a function of time.

Key Concepts

Angular Velocity

Angular velocity is a measure of how quickly an object rotates or revolves relative to another point, expressed in radians per second (rad/s). It describes the rate of change of the angular position of an object and is a vector quantity, having both magnitude and direction. In this problem, the angular velocity is given as a function of time, ω_z(t) = g - bt^2.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, typically expressed in radians per second squared (rad/s²). It indicates how quickly an object is speeding up or slowing down its rotation. Mathematically, it is the derivative of angular velocity with respect to time. In this context, calculating the angular acceleration involves differentiating the given angular velocity function ω_z(t) = g - bt^2 with respect to time.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function, which represents the rate at which a quantity changes. In physics, differentiation is used to determine rates such as velocity and acceleration. For this problem, differentiating the angular velocity function ω_z(t) = g - bt^2 with respect to time will yield the angular acceleration as a function of time.