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Ch 09: Rotation of Rigid Bodies
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Young & Freedman Calc 14th EditionCh 09: Rotation of Rigid Bodies문제 4b
9장, 문제 4b

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

검증된 단계별 안내
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.
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.
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.
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.
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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
가이드 코스
06:18
Intro to Angular Momentum

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.
추천 영상:
가이드 코스
12:12
Conservation of Angular Momentum

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.
추천 영상:
가이드 코스
05:44
Solving Constant and Average Velocity Problems
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