CALC The angle θ through which a disk drive turns is given by θ(t) = a + bt - ct^3, where a, b, and c are constants, t is in seconds, and θ is in radians. When t = 0, θ = p/4 rad and the angular velocity is 2.00 rad/s. When t = 1.50 s, the angular acceleration is 1.25 rad/s^2. (c) What are θ and the angular velocity when the angular acceleration is 3.50 rad/s^2?

Verified step by step guidance

1. Step 1: Differentiate the given function θ(t) = a + bt - ct^3 with respect to time t to find the angular velocity, ω(t). Use the formula ω(t) = dθ/dt.
2. Step 2: Differentiate the angular velocity function ω(t) to find the angular acceleration, α(t). Use the formula α(t) = dω/dt.
3. Step 3: Solve the angular acceleration equation α(t) = 3.50 rad/s^2 for t to find the time at which the angular acceleration is 3.50 rad/s^2.
4. Step 4: Substitute the value of t found in Step 3 back into the original function θ(t) to find the angle θ at that time.
5. Step 5: Substitute the value of t found in Step 3 into the angular velocity function ω(t) to find the angular velocity at the time when the angular acceleration is 3.50 rad/s^2.

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 displacement of the disk drive as a function of time, incorporating constants that affect its motion. Understanding how to interpret and manipulate this function is crucial for determining the position of the disk at any given 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 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 specific times is essential for understanding the motion of the disk drive.

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 second derivative of the angular displacement function. In this scenario, recognizing how to relate angular acceleration to the given function and its derivatives is key to solving for the conditions when the angular acceleration reaches 3.50 rad/s².