A fan blade rotates with angular velocity given by ω_z(t) = g - bt², where g = 5.00 rad/s and b = 0.800 rad/s³. Calculate the instantaneous angular acceleration α_z at t = 3.00 s and the average angular acceleration α_av-z for the time interval t = 0 to t = 3.00 s. How do these two quantities compare? If they are different, why?

Verified step by step guidance

1. Step 1: Understand the problem. The angular velocity ω_z(t) is given as a function of time: ω_z(t) = g - bt^2, where g = 5.00 rad/s and b = 0.800 rad/s^3. We need to calculate two quantities: (1) the instantaneous angular acceleration α_z at t = 3.00 s, and (2) the average angular acceleration α_av-z over the interval t = 0 to t = 3.00 s. Finally, we compare these two values and explain any differences.
2. Step 2: Recall the definition of instantaneous angular acceleration α_z. It is the time derivative of angular velocity ω_z(t). Mathematically, α_z = dω_z/dt. Differentiate ω_z(t) = g - bt^2 with respect to t to find α_z(t). The derivative is: α_z(t) = -2bt.
3. Step 3: Substitute t = 3.00 s into the expression for α_z(t) to find the instantaneous angular acceleration at that specific time. Use the given value of b = 0.800 rad/s^3 in the formula α_z(t) = -2bt.
4. Step 4: Recall the definition of average angular acceleration α_av-z. It is the change in angular velocity Δω_z divided by the time interval Δt. Mathematically, α_av-z = (ω_z(final) - ω_z(initial)) / (t(final) - t(initial)). Calculate ω_z(initial) at t = 0 and ω_z(final) at t = 3.00 s using the given formula ω_z(t) = g - bt^2. Then compute α_av-z using the formula.
5. Step 5: Compare the instantaneous angular acceleration α_z at t = 3.00 s with the average angular acceleration α_av-z over the interval t = 0 to t = 3.00 s. Discuss why they might differ. Instantaneous angular acceleration reflects the rate of change of angular velocity at a specific moment, while average angular acceleration represents the overall change in angular velocity over a time interval. If the angular velocity changes non-linearly (e.g., due to the quadratic term in ω_z(t)), the two values will generally differ.

Key Concepts

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, typically expressed in radians per second (rad/s). In this context, the angular velocity function ω_z(t) describes how the rotation speed of the fan blade changes over time, influenced by constants g and b. Understanding this concept is crucial for calculating both instantaneous and average angular acceleration.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, represented by α_z. It can be calculated as the derivative of the angular velocity function with respect to time. In this problem, finding the instantaneous angular acceleration at a specific time (t = 3.00 s) and the average angular acceleration over a time interval requires a solid grasp of this concept.

Instantaneous vs. Average Acceleration

Instantaneous angular acceleration refers to the angular acceleration at a specific moment in time, while average angular acceleration is calculated over a defined time interval. These two quantities can differ due to the nature of the angular velocity function, which may not be constant. Understanding the distinction between these two types of acceleration is essential for analyzing the results and their implications in the context of the fan blade's motion.