The angular velocity of a process control motor is ω = ( 20 ─ ½ t² ) rad/s, where t is in seconds. Through what angle does the motor turn between t = 0 s and the instant at which it reverses direction?

Verified step by step guidance

1. Step 1: Understand the problem. The angular velocity ω is given as a function of time: ω = 20 - (1/2)t² rad/s. The motor reverses direction when ω = 0. To find the angle θ turned by the motor, we need to integrate the angular velocity over the time interval from t = 0 to the time when ω = 0.
2. Step 2: Determine the time at which the motor reverses direction. Set ω = 0 and solve for t: 0 = 20 - (1/2)t². Rearrange the equation to isolate t²: t² = 40. Solve for t: t = √40 seconds.
3. Step 3: Write the expression for the angle θ turned by the motor. The angle θ is the integral of angular velocity ω with respect to time: θ = ∫ω dt. Substitute ω = 20 - (1/2)t² into the integral: θ = ∫(20 - (1/2)t²) dt.
4. Step 4: Perform the integration. Break the integral into two parts: θ = ∫20 dt - ∫(1/2)t² dt. The first term integrates to 20t, and the second term integrates to -(1/6)t³. Combine these results: θ = 20t - (1/6)t³.
5. Step 5: Evaluate θ over the interval from t = 0 to t = √40. Substitute the limits of integration into the expression for θ: θ = [20(√40) - (1/6)(√40)³] - [20(0) - (1/6)(0)³]. Simplify the expression to find the total angle turned by the motor.

Key Concepts

Angular Velocity

Angular velocity (ω) is a measure of how quickly an object rotates around an axis, expressed in radians per second. In this context, the angular velocity of the motor is given as a function of time, indicating that it changes as time progresses. Understanding angular velocity is crucial for determining how the motor's speed varies and for calculating the total angle turned over a specific time interval.

Angular Displacement

Angular displacement refers to the angle through which an object has rotated about a specific axis, measured in radians. To find the total angle the motor turns, we need to integrate the angular velocity function over the time interval from t = 0 s to the time when the motor reverses direction. This concept is essential for connecting the motor's changing speed to the total angle it covers.

Reversal of Direction

The reversal of direction in a rotating object occurs when its angular velocity changes sign, indicating a transition from clockwise to counterclockwise rotation or vice versa. In this problem, we need to determine the time at which the angular velocity becomes zero (ω = 0) to find when the motor reverses direction. This point is critical for calculating the angle turned during the specified time interval.