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. What are θ and the angular velocity when the angular acceleration is 3.50 rad/s²?

검증된 단계별 안내

Step 1: Start by understanding the given equation for the angle θ(t) = a + bt - ct^3, where a, b, and c are constants. The angular velocity ω(t) is the first derivative of θ(t) with respect to time, and the angular acceleration α(t) is the second derivative of θ(t) with respect to time.

Step 2: Differentiate θ(t) to find ω(t). Using the derivative rules, ω(t) = dθ/dt = b - 3ct^2. Then, differentiate ω(t) to find α(t), which is the angular acceleration: α(t) = dω/dt = -6ct.

Step 3: Use the initial conditions to solve for the constants a, b, and c. When t = 0, θ = π/4 rad, so substitute t = 0 into θ(t) to find a. Similarly, when t = 0, ω = 2.00 rad/s, so substitute t = 0 into ω(t) to find b. Finally, use the condition α(1.50 s) = 1.25 rad/s² to solve for c by substituting t = 1.50 s into α(t).

Step 4: Once the constants a, b, and c are determined, substitute α(t) = 3.50 rad/s² into the angular acceleration equation α(t) = -6ct to solve for the corresponding time t. This will give the time at which the angular acceleration is 3.50 rad/s².

Step 5: Use the calculated time t from Step 4 to find θ and ω. Substitute this value of t into θ(t) to find the angle θ, and into ω(t) to find the angular velocity ω. These will be the values of θ and ω when the angular acceleration is 3.50 rad/s².

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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 rotates about a fixed point, measured in radians. In this context, θ(t) represents the angular displacement of the disk drive as a function of time, incorporating constants that affect its motion. Understanding how to evaluate θ at specific times is crucial for determining the position of the disk drive.

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 can be derived from the angular displacement function by taking its first derivative with respect to time. In this problem, knowing how to calculate angular velocity at different times is essential for understanding the motion of the disk drive.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, measured in radians per second squared (rad/s²). It is obtained by differentiating the angular velocity function. In this scenario, the relationship between angular acceleration and the constants in the angular displacement equation is key to finding the conditions under which the angular acceleration reaches a specified value.