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. (b) What is the angular acceleration when θ = π/4 rad?

Verified step by step guidance

1. 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) and angular acceleration α(t) can be derived by differentiating θ(t) with respect to time.
2. Step 2: Differentiate θ(t) to find the angular velocity ω(t). Using the derivative, ω(t) = dθ/dt = b - 3ct^2. This equation will help us find the angular velocity at any given time.
3. Step 3: Differentiate ω(t) to find the angular acceleration α(t). Using the derivative, α(t) = dω/dt = -6ct. This equation will help us find the angular acceleration at any given time.
4. Step 4: 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, substitute t = 0 into ω(t) to find b. Finally, use the condition that α = 1.25 rad/s² when t = 1.50 s to solve for c.
5. Step 5: Once the constants a, b, and c are determined, substitute θ = π/4 rad into the equation for α(t) = -6ct to find the angular acceleration at that specific angle. This will give the desired result.

Key Concepts

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 position of the disk drive as a function of time, indicating how far it has turned from its initial position. Understanding angular displacement is crucial for analyzing rotational motion and determining other related quantities like angular velocity and acceleration.

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 quickly an object is rotating. In the given 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 angular acceleration at specific points in time.

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 rotational speed of an object is changing. In this scenario, calculating angular acceleration involves differentiating the angular velocity function, which is itself derived from the angular displacement function. This concept is vital for understanding the dynamics of rotational motion and solving for specific conditions in the problem.