The angle θ through which a disk drive turns is given by θ(t) = a + bt - ct³, where a, b, and c are constants, t is in seconds, and θ is in radians. When t = 0, θ = π/4 rad and the angular velocity is 2.00 rad/s. When t = 1.50 s, the angular acceleration is 1.25 rad/s². Find a, b, and c, including their units.

검증된 단계별 안내

Step 1: Start by understanding the given equation for the angle θ(t) = a + bt - ct^3. Here, θ is the angular position in radians, t is time in seconds, and a, b, and c are constants to be determined. The problem provides three conditions: (1) θ(0) = π/4, (2) angular velocity ω(0) = 2.00 rad/s, and (3) angular acceleration α(1.50) = 1.25 rad/s².

Step 2: Use the first condition θ(0) = π/4. Substitute t = 0 into the equation θ(t) = a + bt - ct^3. This simplifies to θ(0) = a. Therefore, a = π/4 rad.

Step 3: Use the second condition, which involves angular velocity. Angular velocity is the first derivative of θ(t) with respect to time, ω(t) = dθ/dt. Differentiate θ(t) = a + bt - ct^3 to get ω(t) = b - 3ct^2. Substitute t = 0 and ω(0) = 2.00 rad/s into this equation: ω(0) = b - 3c(0)^2. This simplifies to b = 2.00 rad/s.

Step 4: Use the third condition, which involves angular acceleration. Angular acceleration is the second derivative of θ(t) with respect to time, α(t) = d²θ/dt². Differentiate ω(t) = b - 3ct^2 to get α(t) = -6ct. Substitute t = 1.50 s and α(1.50) = 1.25 rad/s² into this equation: α(1.50) = -6c(1.50). Solve for c: c = -α(1.50) / (6 × 1.50).

Step 5: Summarize the results. From Step 2, a = π/4 rad. From Step 3, b = 2.00 rad/s. From Step 4, c can be calculated using the formula c = -α(1.50) / (6 × 1.50). Ensure that the units for a, b, and c are consistent: radians for a, rad/s for b, and rad/s³ for c.

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주요 개념

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

Angular Displacement

Angular displacement refers to the angle through which an object has rotated about a specific axis, measured in radians. In the context of the given equation θ(t) = a + bt - ct^3, θ represents the angular displacement as a function of time. Understanding this concept is crucial for analyzing rotational motion and determining how the angle changes over time.

Angular Velocity

Angular velocity is the rate of change of angular displacement with respect to time, typically expressed in radians per second (rad/s). It provides insight into how fast an object is rotating. In the problem, the angular velocity can be derived by differentiating the angular displacement function θ(t) with respect to time, which is essential for finding the constants a, b, and c.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, measured in radians per second squared (rad/s²). It indicates how quickly the angular velocity of an object is changing. In this problem, the angular acceleration is obtained by differentiating the angular velocity function, which is derived from the angular displacement function, and is necessary for solving for the constants in the equation.