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Ch 04: Kinematics in Two Dimensions
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Knight Calc 5th EditionCh 04: Kinematics in Two Dimensions문제 75
4장, 문제 75

A painted tooth on a spinning gear has angular position θ = (6.0 rad/s⁴)t⁴. What is the tooth's angular acceleration at the end of 10 revolutions?

검증된 단계별 안내
1
Step 1: Understand the given angular position equation θ = (6.0 rad/s⁴)t⁴. Here, θ is the angular position as a function of time t, and the coefficient 6.0 rad/s⁴ represents the proportionality constant. Angular acceleration is the second derivative of angular position with respect to time.
Step 2: Differentiate θ with respect to time t to find the angular velocity ω. Using the formula ω = dθ/dt, calculate the first derivative of θ = (6.0 rad/s⁴)t⁴. This gives ω = 4(6.0 rad/s⁴)t³ = 24.0t³ rad/s.
Step 3: Differentiate ω with respect to time t to find the angular acceleration α. Using the formula α = dω/dt, calculate the second derivative of θ. This gives α = d(24.0t³)/dt = 72.0t² rad/s².
Step 4: Determine the time t corresponding to 10 revolutions. One revolution corresponds to an angular displacement of 2π radians. Therefore, 10 revolutions correspond to an angular displacement of θ = 10 × 2π = 20π radians. Solve for t in the equation θ = (6.0 rad/s⁴)t⁴ by substituting θ = 20π. This gives 20π = (6.0 rad/s⁴)t⁴. Solve for t.
Step 5: Substitute the value of t obtained in Step 4 into the angular acceleration formula α = 72.0t² rad/s² to find the angular acceleration at the end of 10 revolutions.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
5m
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주요 개념

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

Angular Position

Angular position describes the orientation of a rotating object in terms of an angle, typically measured in radians. In this question, the angular position θ is given as a function of time, indicating how it changes as the gear spins. Understanding this concept is crucial for determining how the gear's rotation evolves over time.
추천 영상:
가이드 코스
14:03
Rotational Position & Displacement

Angular Velocity

Angular velocity is the rate of change of angular position with respect to time, usually expressed in radians per second (rad/s). It can be derived from the angular position function by taking its first derivative. In this case, knowing the angular velocity at a specific time will help in calculating the angular acceleration.
추천 영상:
가이드 코스
06:18
Intro to Angular Momentum

Angular Acceleration

Angular acceleration is the rate of change of angular velocity with respect to time, measured in radians per second squared (rad/s²). It can be found by taking the derivative of the angular velocity. In this problem, calculating the angular acceleration at the end of 10 revolutions requires understanding how the angular position function relates to both angular velocity and acceleration.
추천 영상:
가이드 코스
12:12
Conservation of Angular Momentum
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